Time is money. So is your job. With both to maintain, travelling had to be made less time consuming. We tailor our itineraries in a way that the distance between two cities in opposite hemispheres of the country is traversed in lesser time than it takes from Behala to Barrackpore or Colaba to Kandivilli. In the last one year, all I can think of is, two train journeys…compared to innumerable aeroplane rides I had. Two hours flight replaced by 2 whole days in the train is unthinkable. However, Mr. Perfectly Alright and I are ceaselessly in love with train journeys and sometimes make it a point we travel by train. Needless to say, the excitement in travelling by a train is incomparable with any other journey (I haven't tried space-ships though).
The thrill begins when you have a Waiting list or an RAC ticket. Every now and then you track it, with the anticipation that someone's tickets are cancelled to give way to yours. You apply all your means, connections and bribes to get the tickets done. Especially, for Bengalis, travelling is like politics, adda and Uttam Kumar. Imbibed in their neurons. Life without travelling is equivalent to no life at all. (You see tourists dying in bomb-blasts in Kashmir or land-slides in Kumayun. All of them are Bengalis).
And once your ticket is confirmed, starts the real fun. Packing! There is no 20 kilos weight limit. You can almost carry your 800 sq.ft. home into the train with slight variations. Safety-pins, idols of Ram-Sita, gas burners, packets of puffed rice and Maggi, pickles, blankets, radio sets, pets (non-barking ones) and a range of other useless and useful things tuck themselves conveniently in those huge metal trunks and trolleys (read 'suitcase'). In the event they cannot tuck themselves so conveniently, children would be asked to sit on the upper-lid, so that the luggage can be closed properly and locked tightly. And, if there is a numeric lock, then each time the luggage is opened (or closed), the person turns a Vinay Pathak in Bheja Fry! (You know what I mean).
One thing I fail to notice nowadays, which was very common when we were small. The owner of the luggage would have stickers bearing his/her initials fixed on the suitcase on two sides of the handle (there was a specific space for doing it). In yester years, suitcases used to come with alphabet-stickers. You choose your initials, stick them and when the luggage is lost, the whole world would come to know that it belonged to you by a mere B.G, C.N or S.P (as if)!!!!
Once things are packed, starts the first round of counting the luggage while putting them in the boot of a taxi (fondly called "dicky") and then see for the umpteenth time whether the cab-driver has locked it properly. On the way to the station, at every red signal where the taxi has stopped, check whether the dicky is still locked. The second round of counting starts after reaching the station. The third round follows while handing over the luggage to the Coolie and the final one once you've boarded the train. The in-between mental counting goes on while following the Coolie till the train. And of course, keeping a count equally applies to little children, if you are travelling with a number of them (except the dicky and the coolie-part, of course).
For Bongs in Kolkata, like us, I think you would agree, the delight of travelling begins at the very step onto the Howrah Bridge. It is like a lovely trailor to a lovelier movie that lies ahead (unlike the present Yash-Chopra movies). The thrill, the excitement and the boundless joy begins just at the visibility of Ganga, the mightiness of the suspension bridge and the Howrah Station that beckons. And then, the sight, smell and sound of the station make you ecstatic..instantly. You don't mind the sweat, the loud bustle, the annoying pleadings of hawkers and the relentless bargaining sessions with Coolies.
One more thing I have noticed. Whenever we used to travel in big groups, people would join each other near the "Boro-ghori" (the big clock) in Howrah Station. This was one unfailing landmark for all, I guess. Not only for travellers, it is the greatest rendezvous point for lovers, friends, relatives as well, before they take their respective trains. In the age of no-mobile phones, people who got lost in the railway station, would invariably wait near the "boro-ghori" with the anticipation that their friends or relatives would come here atleast once to look for them. I have heard, office-goers check if their wristwatches match the hands of the big clock during 'awpish'-time.
And then starts the advent towards the train once the platform is declared. Checking names on passengers' list (complaining "hey I am 36 years only, they have written 40 against my name"), pushing way thorough mounds of luggage, passengers and porters, dealing with the one who carried yours and ending the bargain with likes of "tang maat karo, tang maat karo, kafi ho gaya, zyada ho giya.." (in sonar kella-style). Achha, have you noticed, we, Bongs always speak to Taxi-drivers and Coolies in Hindi? Why???
And now that your train has started, you have a mixed feeling of fear and frolic. You cross-check the tickets and in case you have a 15 years old Mumpi traveling with a 26 years old Sumana's ticket (due to last moment plan-change), you teach her to settle in the upper berth as soon as the ticket-checker advents, so that he fails to fathom her exact age from a distance.
The beginning of the journey is characterised by men putting their sneakers into their bags and pulling out from newspaper-wraps their rubber sandals (which we Bongs affectionately call Hawai-choti), taking off their trousers and slipping into a pair of shorts (also known as haap-paant) or lungi. Yes, Indian men often have a separate set of clothes, one for boarding the train and the other for travelling in it. It is then followed by expert comments regarding the cleanliness of the toilet and pleading co-passengers to use it to their heart's content with the anticipation that the latrines would soon become soiled and unfit for human defecation.
As the train moves on, crossing one state after the other, conversations with neighbouring passengers intensify. We soon get to know their whole itineraries including the money invested in the tour. Their aloo-parathas are exchanged with luchis during dinner and upper berths exchanged for lower ones, pleading arthritis (telephone numbers are exchanged too!). Swapping seats with friendly co-passengers is a widely accepted phenomenon. I remember, many years ago, while travelling from Chennai to Kolkata, a couple with 2 kids had waiting list tickets. (In Sleeper class, you are allowed only to board the train with such tickets but NOT to sit). My father had made me sleep with one kid while the other slept with my brother. My parents left their entire berth to the couple. The two sets of parents had kept awake the entire night sitting and talking to each other.
And no journey however small or big, is complete without the mention of food. The moment one boards the train, I guess, the appetite multiplies infinitely. Tea is had at an interval of 10 minutes, followed by 'coal-dinks', jhaal-muri, roasted peanuts, pakora, cutlet and bread-omelette (the vendors have a peculiar way of saying it!). When we were small, the second meal in the train was always from the pantry (the first being home-food). And the enthusiasm of eating train-food was like having a lunch at a 5-star hotel and the excitement as great as turning 18. The pickle, the dahi, the almost-always-arhaar daal and the presence of 2 eggs in the egg-curry (when you had expected one) was heavenly, trust me. It was a feast for eyes and for the stomach.
Train food was punctuated with food from big junctions. Kharagpur signified puri and alu-sabzi (unskinned potatoes, i.e), Bhubaneshwar stood for sweets, bananas and filling of water-bottles, Vizag and Vijaywara for fruit juices, Nagpur for oranges. The stations in South India would invariably distinguish themselves with Vadas (which looked like doughnuts), Dosas, Curd Rice, Egg-Biriyani and onion-raita, all wrapped in Sal leaves.
Filling water bottles in railway stations reminds me how anxious we used to get at the thought that the train may start any moment and leave the station with Baba being stranded there forever. (To be honest, we, as kids secretly wanted this to happen atleast once, so that we could pull the chain. That would have been a lifetime experience).
Such simply exciting and excitingly simple were the train rides. In one of my Kolkata-to-Mumbai flights, at the time of getting down from the aircraft, my neighbour asked me "Mumbai?" I felt like saying "No. Uzbekistan". In the entire 2 and a half hours flight, I was feeling terribly bored and was waiting to talk to someone, while the man on my next seat had his eyes glued to an India Today. Now that the flight has come to an end, he finds the time to talk! Had it been a train journey, I would have made him play cards and/or antakshari with me, share his lunch and books and knew enough of him to write a short biography. Exaggerations apart, what I mean to say is that the warmth and the joie de vivre a train journey offers are truly unmatched.
The last time I travelled in a train, I made it a point to collect as much memories I could. I had as much pantry-food as possible, IRCTC tea was had in abundance accompanied by bread-omelette, ludo bought and played but most importantly, soap papers bought in a bundle. This is one thing I miss so much. With the advent of wet wipes and hand sanitisers, that is. When we were kids, wedding feasts were followed by finger bowls (colourful plastic ones) with soap papers in each (instead of lemons, as it is today). And these soap papers were a MUST in train journeys as well.
Well, as I said, in our endeavours to maintain time and jobs, travelling cannot be made a prolonged affair. Flights have thus replaced train journeys in a major way. Simple ways of life has been swapped with faster and more refined ones. Nevertheless, for people like us, whose childhood memories are overflowing with moments spent in the train (sleeper class, to be noted), no journey can ever replace them in terms of the fun had. Hearts are trained that way!!
totally get that banhea di. the love for train rides and the sadness of missing them so badly made me come up with this post. am eternally in love with trains..especially sleeper classes (minus the latrines, of course).
little can be added when a word-smith such as yourself chooses to present their ideas with such logical-'n-Lawyer-ish (definitely NOT lawarisssss) precision!!
o god Kanko, you spoke my heart!!! of cors, the luchi-mangsho part was one of the greatest attractions. As, I have written here the first meal would always be home-food. Initially my parents would only take luchi-aloordam, contemplating non-veg would be messy. But the "aar-chokhe-lobh-dewa" on non-veg food eaten by co-paasengers were soon taken care of by replacing aloor dum with kosha mangsho..my mouth is becoming niagra falls while typing this.
And readers, what totally missed my memory is "folano balish" or air-pillows!!how could i forget those? the fun of blowing air into it, getiing shoulder-aches after sleeping on them and while pushing the air out of them into someone's face or ears..i so miss dat too!!
"Had it been a train journey, I would have made him play cards and/or antakshari with me, share his lunch and books and knew enough of him to write a short biography"...ha ha...lucky guy!
"The stations in South India"...how can all the four states be all typecast as one?
:( I didinot generalise at all. What I mentioned was the fooditems available in the stations down south. I have extensively travelled down south and many a times. And whatever i could recollect from my small memory. Not typecasting, please.
:( Didnt get your name. Mention it please.
Thіs post was helpful. I actually really enjoyеd reaԁing it.
not thе best thing I have еver donе but regaгdlesѕ thаnks!


What are you doing every Saturday and Sunday afternoon, the first three weeks of December? If you've got nothing scheduled, why not pick one weekend and head to the Butterfly House in Chesterfield? For a limited time only, kids one year's old and younger are welcome to enjoy Supper with Santa from 4:30 p.m. until 7 p.m. Older children are also welcome. However, everyone over one year of age must pay an admission fee to get in and break bread with the big guy. Advance registration is required to attend dinner with the jolly old elf and it may be completed online.
When you're done dining with the North Pole's top celebrity, consider taking a stroll through the Annual Garden Glow event in St. Louis. It's an absolutely amazing array of family friendly, holiday lights displays, Christmas trees, festive music and seasonal fare. The event opened up on November 22nd and is expected to run through Saturday, January 3rd, from 5:30 p.m. until 9:30 p.m. on designated days. Advance tickets are a must and prices start at $6 per person.
If the kids love G-scale trains and miniature Christmas scenes, make plans to attend the venue's Gardenland Express too. It's a combination event that should appeal to moms, dads and kids alike. Parents will have opportunities to gaze at all sorts of live holiday finery while the kids can check out the trains. The event is slated to take place daily from 9 a.m. until 4 p.m. from now until Sunday, January 4th.
The older kids may also enjoy candle making at the Shaw Nature Reserve in nearby Gray Summit. It's set to take place twice on Saturday, December 13th at the reserve's Adlyne Freund Center. The first candle making session is due to kick off at 9:30 a.m. sharp and run until 11:30 a.m. The second session is anticipated to begin at 1 p.m. and end at 3 p.m. All candle making supplies will be provided by the reserve's staff and admission fees start at $3 per person. Advance registration is strongly suggested so the kids don't get shut out by the holiday crowds.
Those are just a few of the many holiday events taking place at the Missouri Botanical Gardens and the surrounding area. To learn more about the top seasonal activities St. Louis has to offer, please contact us today. We can help you find the best places to get and stay merry all year round.


Timothy Murphy School (TMS) in San Rafael, CA is currently looking for a post-degree, pre-licensed therapist with a PhD or PsyD in psychology to join our clinical team as a Registered Psychologist. Clinical supervision will be provided by a licensed psychologist on-site and all clinical hours may be applied towards licensure as a psychologist in California. This will be a part-time position, working from 8:30-3:30 four days of the week, for an average of 28 hours per week. The position begins on June 19, 2017 and a monthly stipend is available.
Timothy Murphy School is a nonsectarian non-public school for students who struggle to access their education due to significant emotional disturbances and behavioral challenges. The majority of our students are placed due to symptoms deriving from early childhood trauma and disruption of the attachment system. TMS clinicians therefore need to be familiar with the psychology and treatment of these conditions.
Our program is built upon the ARC system (Attachment, Regulation, and Competency), with the book Treating Traumatic Stress in Children and Families (Blaustein & Kinniburgh, 2010) acting as our core treatment manual. The ARC model informs individual treatment, our social-emotional competency groups, and the school's general behavior support program. The Registered Psychologist will be expected to become highly familiar with the ARC system and to provide clinical interventions based on its principles.
The Registered Psychologist plays an important role in the TMS clinical team and the school as a whole. This position provides a broad range of opportunities for both clinical experience and professional development.
Provide psychotherapy to individual students between the ages of 8 and 18.
Caseload will hover around 7-9 clients.
Co-facilitate our bi-weekly social-emotional competency groups.
Provide delegated supervision to an MFT trainee.
Develop and present psychoeducational workshops for all TMS staff.
Generate psychoeducational reports and Behavior Intervention Plans.
Provide consultation services to school staff and parents/guardians.
Participate in Individualized Education Program (IEP) meetings.
Other duties as assigned by the clinical supervisor or school director.
Awarded PhD or PsyD from an APA-accredited graduate program in clinical psychology.
Experience providing individual mental health services to children, preferably with a focus on trauma and attachment-based issues.
Experience with psychological testing, preferably with children.
The capacity to remain calm, focused, and professional when faced with unexpected behavioral challenges.
Familiarity with the principles of operant conditioning as they apply to human behavior.
Experience working within a school environment is a plus.
Applicants will need to submit a CV, proof of graduation (e.g. informal graduate transcripts or diploma), three letters of recommendation, a de-identified psychological assessment report, and a letter of interest. Please mail your application to Dr. Ash Bowie at Timothy Murphy School, 1 St. Vincent Drive, San Rafael, CA, 94903. Or you may email the documents to ashb@timothymurphyschool.org.


The caves at Sherishitti were used by the Adoo to store a vast collection of arms and ammunition. Troops of the Sultan's forces and SAS Firqats launched an attack to capture the caves in January 1975. The operation was mounted from one of the government outposts at Defa, known as 'The position at Defa'. The position was constantly attacked through out the war and was called upon often to give artillery fire support to other operations. At one stage over 2,000 rounds were fired from the position in one week.
The fighting in the wadi, in front of the caves in the dense undergrowth, was a hard fought and bitter match against a determined Adoo. On the second day of the fighting, a company of the Sultan's forces attempted to cross some open ground in the wadi, despite their commanding officer being told by the SAS that it was a bad idea. The Adoo could not believe their luck and easily cut down the advancing troops as they tried to cross the wadi. In less than an hour the casualties amounted to 13 dead and 22 wounded. For the rest of the day the SAS were forced to take on the Adoo single handed to restore some fighting spirit in their allies. The SAS poured machine gun and mortar fire into the Adoo positions and provided covering fire for the survivors of the Sultan's forces. One SAS Fijian trooper carried out a wounded Arab under fire after the man had been dropped three times by his fleeing comrades.
The caves did not fall into government hands until after the war but the Sultan's forces denied the Adoo from using them again by bombarding anything that moved near the caves from positions on the high ground taken on the first day of fighting.
In September of the same year, an SAS patrol was sent out from 'The position at Defa' to neutralize an Adoo Katyusha (122mm Soviet made missile) that the Adoo were firing at Defa. The SAS patrol made contact with the Adoo and a fierce firefight ensued. Lance Corporal Geordie Small was hit and bled to death before he could be casevaced out. Lance Corporal Tony Flemming was also hit with the bullet severing his spinal column. A third SAS solider was hit with the bullet passing through his neck. The SAS broke contact with the Adoo knowing that they had given as good as they got.


Shop cake pop sticks at Bakers Party Shop. We carry bulk cake pop sticks, clear cake pop sticks, paper cake pop sticks, metallic cake pop sticks, plastic cake pop sticks and patterned cake pop sticks. Whether you are looking for a basic cake pop stick so your sweet work shines or something a little more elegant to go with your party or event we have something for you!
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text:
John Challis (I)
Actor | Writer | Soundtrack
John Challis is an accomplished character actor, personality and raconteur whose wide-ranging career has gone from classical roles on the stage to heavies in television drama, a multitude of ... read full biography
Known for Only Fools and Horses (1981-2003), Doctor Who (1976), Z Cars (1967-1975), Open All Hours (1976)
Aug 16, 1942 (age 76) | 6' 1¾" (1.87m)
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redpajama_set_name: RedPajamaCommonCrawl


In general, work/life balance for me is not an evenly dispersed kind of thing but more like all-consuming-projects punctuated with an annual travel adventure when I disappear overseas for four weeks. I used to want to balance it all out, but now I accept that I actually like life being 'all on' when I'm hooked on a project and then changing track and escaping somewhere else when it's done. Something I could do better is managing my wellbeing across these times. I tend to burn out, thinking I'll sleep/breathe/recharge later. Well… I really don't really want to sleep/breathe/recharge once or twice a year. I'd rather that be an all of the year thing.
So I started looking after myself better. I cut down on chocolate (I haven't eaten any since the 28th June 2015 actually). I didn't let my morning bush walk slide off my schedule when I was busy, it has become a non-negotiable. I started a regular weekly yoga practice after years of patchy attendance, and I sat down to daily meditation. I've kept to it all pretty faithfully and the result was that at the end of a pretty big work project last year I needed some time out for sure, but I wasn't an exhausted shell left barely standing. I had energy, so after my work trip to Singapore I hopped on a plane to Koh Samui and straight to Vikasa Yoga Retreat. I stayed for 15 days.
I found Vikasa through a Google search, it sounded promising with great Trip Advisor reviews and plenty of travel blog trip reports.
Skiing was a big deal for me as I have only done it once in my lifetime, and it was on the disaster scale. I was seventeen, on a student trip to the USA west coast on the G'day USA program. We went to the snow somewhere in the extreme north west of the country, I can't even remember the name, it was south of the Canadian border. Our group instructor seemed bored and not that bothered. He gave us a half-hearted lesson before putting us all on the chairlift up to the top (it seemed like a ginormous mountain, but I concede that time may have increased the dramatic scale in my memory). The chairlift was fun, but when we got to the top I didn't know what to do! I put my feet down, slid off the chair and then kept sliding. Down. The. Mountain. The good news is that I didn't crash. The bad news is that I didn't know how to turn or control my speed at all, so I picked up speed until I was literally airborne for half of the descent. When I neared the bottom I saw that it leveled out to end in an excavation pit where there were trucks and bulldozers (again, memory scale maybe be slightly distorted). So to avoid skiing off the cliff, I had to throw myself sideways, where I cut my cheek on the ice. Epic ski fail.
Cut to 15 years later and I thought I'd give it another go.
Because I was on my own I visited Snow Biz the afternoon before to book in a gear hire beginner package which included ski and clothing hire, lessons and bus transport up the mountain. I rocked up very early the next morning to get fitted for skis, pants and jacket then was one of the first to get in line for the buses, which depart right outside the shop. There are two options – The Remarkables and Coronet. I did some googling which would be best for a beginner and decided to go to The Remarkables.
The bus ride was easy, though pretty intense considering the steep, winding climb on a narrow, snowy road! We arrived and I felt a bit out of the loop on what/how. Everyone was there in friend or family groups and seemed to know what they were doing. The intrepid solo traveller is not immune to occasional pangs for the comfort and ease of being with friends. But the only option is to brush it aside and make it work. I watched for awhile and then changed into my ski boots there in the car park and then trudged up to the people-conveyor-belts (what are they called?) and to the building with everyone else.
I figured it out from there – I put my bag in the storage room, leaned my skis on one of the numbered stands, went to the cafeteria for a coffee, snapped some photos and then it was time for my first lesson!
Luckily there were some other solo peeps in my group so we quickly forged a bond of support to keep each other's morale up and clapped and cheered our gradual improvement. Our instructor, a twenty-something Welsh guy, was absolutely wonderful. Funny, kind and understanding – he played a huge part in building my confidence. We had a two hour morning lesson, stopped for a lunch break and then had an afternoon two hour lesson.
I couldn't believe that by the end of the day I was capable of skiing down the beginner slope, controlling my direction and speed and most importantly STAYING UPRIGHT!


text:
Write me: rachel.howard@gmail.com
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Black Choreographers Back on the Move
HomeBlogBlack Choreographers Back on the Move
January 29, 2005 Rachel 0 Comments
In the Sunday Chronicle, I talk to the founders of the new Black Choreographers Festival, which models itself on the Black Choreographers Moving Towards the 21st Century fest that died out about a decade ago:
?The year 1995 was a landmark for the Black Choreographers Moving Toward the 21st Century Festival, or BCM, as insiders know it. That year, the showcase — which sparked a national dialogue in the African American dance world with its start in 1989 — featured all Bay Area artists. A 27-year-old virtuoso named Robert Henry Johnson held the house in rapture, fluttering between ballet, hip-hop and jazz moves with the delicacy of a butterfly and the brashness of a boxer. Another still-young talent, Robert Moses, danced a solo wearing a collar and a face full of rage, provoking tears and exclamations.
Three women in particular were watching those performances with hope and excitement. Laura Elaine Ellis made her choreographic debut in that festival. Kendra Kimbrough Barnes and Shereel Washington were still students at San Francisco State. "It was life-changing for me," Ellis says one recent morning over coffee, her large eyes widening. "The synergy was just amazing."
Kimbrough Barnes nods her head full of braids toward the tall, stately Washington. "It was so thrilling for us to have a place to look forward to going when we finished with college. It was inspiring."
But the next year, the festival was gone."
O's Little Guide to Starting Over
The Autobiographer's Handbook: The 826 National Guide to Writing Your Memoir
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Typing and canceling, retyping and canceling again.. why is so difficult to describe beautiful things? Maybe I'm just better to express myself with photos, rather than words. Or maybe sometimes words just aren't good enough to describe what our eyes can see. Tuscany is a representative of art in the world in all the aspects. Sculpture and architectural masterpieces, extraordinary paintings and lanscapes followed by good Chianti, gorgeous food and unique Tuscan dialect, which I simply love. I've visited almost all the Tuscan localities designed as World Heritage Site by UNESCO.. Florence, Siena, Pisa, San Giminiano, Arezzo, Cortona, Lucca.. every time I go back, I'm excited like it's my first time.
Tuscany could be easily compared to a perfect wedding. What you need to do is replace the landscapes and art, with friends and family. Renske and Marc were representing Tuscany, but what made their wedding special, were those landscapes and art called friends and family. It was so inspiring watching the harmony between so many people. It was magic. Thanks for such experience!
I also want to thank Aleks Kus, the best assistant possible a photographer could imagine! Never tired with a huge pocket of passion for his work, he was standing shoulder to shoulder with me on this 2 days adventure.
By the time I'm preparing the wedding post, enjoy watching the 'Day One' of the wedding in Tuscany.


text:
\section{Introduction}

{Next-generation large-scale stucture (LSS) surveys such as  Euclid~\cite{Amendola:2016saw}, LSST~\cite{Zhan:2017uwu} and the SKA~\cite{Maartens:2015mra}, together with cross-correlations between these surveys~\cite{Fonseca:2015laa,Alonso:2015sfa}, will probe the distribution of galaxies on ultra-large scales (above the equality scale), where the effects of general relativity (GR) can become important.} It is therefore imperative that the theoretical model for these tracers is formulated consistently in GR.

We do not yet {fully} understand how galaxies form from the initial curvature perturbation and then evolve under a given theory of gravity to become what we observe today,  forcing us to adopt an effective field theory-like approach to model their large-scale behaviour. This approach uses perturbation theory techniques to model  tracers as a function of the long-wavelength mode of a set of physical operators, while averaging over the short-wavelength mode component within a local patch. The averaged contribution of the short modes is then incorporated as bias parameters, which may be determined from observations or  N-body simulations. The goal here is to be able to describe the observed statistics of any tracer with as many bias parameters as may be required within a range of scales where the perturbation theory description may be trusted. The bias parameters appear as coefficients of the physical operators $\mathcal{O}(\tau, {\bm{x}})$ in a perturbation theory expansion of the tracer proper number density contrast
\begin{eqnarray}\label{eq:generalbiaspres}
\delta_{\rm{g} }(\tau ,{\bm{x}})=\sum_{n}b_{\mathcal{O}^{n}}(\tau) \{\mathcal{O}(\tau ,{\bm{x}})\}^{n}\,,
\end{eqnarray}
where $\delta_{\rm{g}} = n_{\rm{g}}/\bar{{n}}_{\rm{g} }-1$ is the density contrast of a particular galaxy type, $b_{\mathcal{O}}$ is the bias parameter, $n_{\rm{g}}$ is the galaxy number density and $\bar{n}_{\rm{g}}$  is the mean. {$\mathcal{O}(\tau ,{\bm{x}})$  consists of a set of operators that may be constructed from the irreducible decomposition of higher than one spatial derivative of the initial curvature perturbations~\cite{McDonald:2009dh}.} The crucial result is that the evolution over long time scales, which is natural in the case of LSS tracers, can be dealt with order by order in perturbation theory \cite{Senatore:2014eva,Mirbabayi:2014zca}. That is, even though the formation history is highly nonlocal in time, the bias expansion can be written as local in time, as in
equation \eqref{eq:generalbiaspres}. Whether the expansion is performed at initial time (Lagrangian) or at final time (Eulerian) is then a matter of choice. More generally, any complete linearly independent combination of these observables at any order in perturbation theory leads to an equivalent bias expansion~\cite{McDonald:2009dh,Desjacques:2016bnm}.

An open problem in the effective description of biased tracers is to determine the number of distinctive operators that need to be included for an accurate description of galaxy clustering at any given order in perturbation theory. Within  the Newtonian approximation, it is now well understood what key operators are to be included and also how to construct them from the initial curvature perturbations \cite{Desjacques:2016bnm}. This is straightforward in the Newtonian approximation mainly because the energy constraint equation, Poisson's equation, which relates the initial curvature perturbation to the matter density contrast, is linear and there is a unique Eulerian frame in which the initial density field is related to the evolved density field. On {ultra-large} scales, we need to apply GR, and its energy constraint equation leads to a nonlinear relationship between the curvature perturbation and the matter density field. In addition, the Eulerian frame is not unique in GR, hence the gauge choice becomes an issue as well.

{{Galaxy bias in GR was first discussed in \cite{Challinor:2011bk,Bruni:2011ta,Baldauf:2011bh,Jeong:2011as} but at the linear order.  The importance of expressing the galaxy bias model in local coordinates was discussed in \cite{Baldauf:2011bh,dePutter:2015vga}.
The  comoving-synchronous gauge  was identified as a unique gauge choice in a Lagrangian frame for specifying the initial galaxy bias in  GR at second order in \cite{Bertacca:2015mca}. The details on  how to construct a consistent local coordinate valid on horizon scale up to second order in perturbation theory were discussed in \cite{Dai:2015jaa}. There is currently no study that  brings these pieces of information together to give a  consistent expression for the local galaxy bias model in GR at second order.  This is a gap we aim to fill.}}

{The main purpose of this paper is to provide a derivation of the local galaxy bias model (including tidal stress) within GR at second order in perturbation theory in a universe dominated by dust plus a cosmological constant,
  for Gaussian initial conditions. We also investigate how GR effects influence galaxy clustering.}

The structure of this paper is as follows. In section II, we provide a formula for the conservation of galaxy number in the comoving-synchronous gauge, which is the unique Lagrangian frame in GR \cite{Bertacca:2015mca}.
In section III, we introduce the local coordinates at initial time where the initial galaxy density is related to a set of local operators constructed from the initial curvature perturbation. Using the conservation of galaxy number,  we relate the initial galaxy density  to the evolved galaxy density. We show that distortion of the Lagrangian volume leads to a GR correction to the galaxy bias model at second order. In section IV, we show how to relate the galaxy bias model in comoving-synchronous gauge (C-gauge) to various Eulerian gauges such as the total matter (or comoving orthogonal) gauge  (T-gauge) \cite{Villa:2015ppa},  Poisson gauge (P-gauge), N-body gauge \cite{Fidler:2017pnb}, N-Boisson gauge \cite{Fidler:2018geb}, and other possible Eulerian gauges.

{\bf{Notation}:} {Greek letters denote space-time  indices and Latin denote spatial indices. The (averaged) 4-velocity field of galaxies/ matter is $u^\mu=dx^\mu/d\tau$, and the world-lines are labelled by the comoving coordinates $\bm{q}$. The matter density contrast is $\delta_{\rm m}= \delta \rho_{\rm m}/\bar{\rho}_{\rm m}$ and the galaxy number density contrast is $\delta_{\rm g}= \delta n_{\rm g}/\bar{n}_{\rm g}$.
We expand perturbations up to second order as
  $ \delta = \delta^{\text{\tiny ({{1}})}} +  \delta^{\text{\tiny ({{2}})}} /2$. }

\section{Conservation of galaxy number  in general relativity}

\begin{figure}[!h]
\includegraphics[width=80mm] {Worldline_Gaussian2.png}
\caption{We assume that galaxies form  around peaks (see figure \ref{fig:patches} for details)  of the matter density field {at $\tau_{\rm{ini}}$ and evolve to $\tau$ in accordance  with a set of conservation and propagation equations.
A $\tau=\,$constant  hypersurface orthogonal to $u^{\mu}$ is shown. } }
\label{fig:Geodesiceqn}
\end{figure}
We show in figure \ref{fig:Geodesiceqn} the conceptual set-up for the formation and  evolution of galaxies that we describe as biased tracers of the dark matter density field. In this set-up,  galaxies form within a patch of radius $R$ and evolve under the influence of gravity until observed at proper time $\tau$.

Firstly, we derive the relation between  galaxy densities at $\tau_{\rm{ini}}$ and $\tau$ in C-gauge. The C-gauge is a unique Lagrangian frame for irrotational dust  fluid in GR. The line element in C-gauge is
\begin{equation}
ds^2 = - d\tau^2 + a^2(\tau) \gamma_{i j}(\tau, \bm{q}) dq^i dq^j\,,
\end{equation}
where  $a(\tau)$ is the scale factor and $\gamma_{i j}$ is the conformal metric on hypersurfaces orthogonal to $u^\mu$.
The conservation {of galaxy number  within a small volume $dV_C$ in GR} is given by,
\begin{eqnarray}\label{eq:GRconservation}
{dN=n_{gC }(\tau,\bm{q})dV_C(\tau,\bm{q})=n_{gC }(\tau_{\rm{ini}},\bm{q})dV_C(\tau_{\rm{ini}},\bm{q})\,, \qquad dV_C=a^3\sqrt{\gamma}\,d^3q\,.}
\end{eqnarray}
Using $n_{\rm{g C}} = \bar{n}_{\rm{g}}(1 + \delta _{\rm{g C}})$, equation \eqref{eq:GRconservation} implies
\begin{equation}
1 + \delta_{{\rm{gC}}}( {\tau, \bm{q}}) = \big[1 +\delta_{\rm{g}}^{\rm{L
}}({\bm{q}}) \big]\, \frac{\sqrt{\gamma^{{\rm{ini}}}({{\bm{q}}})}}{\sqrt{\gamma(\tau,{{\bm{q}}})}}\,.
\label{eq:deltagc}
\end{equation}
Here ${\delta_{{\rm{gC}}}=\delta n_{\rm{gC}}/\bar{n}_{\rm g}}$ is the  galaxy density contrast  in C-gauge, $\gamma^{\rm{ini}}_{ij}({\bm{q}})\equiv \gamma_{ij}(\tau_{\rm{ini}},{\bm{q}})$ is the `seed' metric and  $\delta_{\rm{g}}^{\rm{L
}}({\bm{q}}) \equiv \delta_{\rm{gC }}(\tau_{\rm{ini}},{\bm{q}})$ is the initial (Lagrangian) galaxy density contrast.

The evolved metric can be written in terms of the displacement field $\beta^i$ as  \cite{Rampf:2014mga}
\begin{equation}\label{eq:metric}
 a^2(\tau)\gamma_{ij}(\tau,{{\bm{q}}}) dq^i dq^j =  a^2(\tau)
\big[1 - 2 \mathcal{B}(\tau,{{\bm{q}}}) \big]  \gamma^{ {\rm{ini}}}_{kl} ({{\bm{q}}})
\big[ \delta^k_i +\partial_i\beta^k (\tau,{{\bm{q}}}) \big]
\big[ {\delta^l_j} +\partial_j\beta^l (\tau,{{\bm{q}}}) \big]
dq^i dq^j\,,
\end{equation}
where $ \mathcal{B}$ is the trace of the scalar perturbations of $\gamma_{ij}$~{(see Appendix \ref{sec:CosmoPert} for details)}.
In the limit {$\mathcal{B}\to 0$},
\begin{equation}
 \frac{\sqrt{\gamma^{{\rm{ini}}}({{\bm{q}}})}}{\sqrt{\gamma(\tau,{{\bm{q}}})}}
 ~~  {\to} ~~ J (\tau, \bm{q}) \equiv \det \big[\delta^k_i +\partial_i\beta^k(\tau,{{\bm{q}}}) \big]\,,
\end{equation}
we reproduce the Newtonian {approximation
\begin{eqnarray}
1 + \delta_{{\rm{gC}}}( {\tau, \bm{q}}) = \big[1 +\delta_{\rm{g}}^{\rm{L
}}({\bm{q}}) \big]\,J( {\tau, \bm{q}})\,\qquad (\mathcal{B}=0)\,,
\end{eqnarray}
to the full GR expression \eqref{eq:deltagc}.}
The seed metric is given by \cite{Bruni:2014xma,Rampf:2014mga}
\begin{eqnarray}
\gamma_{  ij}^{\rm{ini}} = \delta_{ij}\,{\exp}\left[-\frac{10}{3} \Phi(\tau_{\rm{ini}},{{\bm{q}}})\right]=\delta_{ij}\left[1 - \frac{10}{3} \Phi(\tau_{\rm{ini}},{{\bm{q}}})+ \frac{50}{9}\big(\Phi(\tau_{\rm{ini}},{{\bm{q}}})\big)^2\right]\,,
\label{eq:initialmetric}
\end{eqnarray}
where $ {\Phi^{\rm{ini}}}({\bm{q}}) \equiv \Phi(\tau_{\rm{ini}},{\bm{q}})$ is  related to the initial curvature perturbation $\zeta = -{5} \Phi^{\rm{ini}}/3$.
Assuming that the initial matter density field is nearly smooth~\cite{Matsubara:2008wx}, the conservation of the matter density field implies that
\begin{equation}\label{eq:ratioofdeterminants}
1+\delta_{\rm{mC}}(\tau,\bm{q}) = \frac{\sqrt{\gamma^{{\rm{ini}}}({{\bm{q}}})}}{\sqrt{\gamma(\tau,{{\bm{q}}})}} \,.
\end{equation}
Note that in C-gauge there is a residual spatial gauge freedom, $q^i \rightarrow q^i + \xi^i (q^k)$, corresponding to how one assigns coordinates $q^i$ to the
fluid world-lines. This residual gauge does not affect
$\sqrt{\gamma^{\rm{ini}}}/{\sqrt{\gamma} }$, thus the evolved matter density is
invariant under this residual spatial gauge transformation.
{Using  equation \eqref{eq:ratioofdeterminants}, the GR conservation equation   \eqref{eq:deltagc} can be rewritten as}
\begin{eqnarray}\label{eq:conservationgen}
1+ \delta_{\rm{gC}}(\tau,{ \bm{q}})& =&
\left[ 1 +\delta_{\rm{g}}^{\rm{L
}}({\bm{q}})\right]\left[1 + \delta_{\rm{mC}}(\tau, {\bm{q}})\right]\,.
\end{eqnarray}
{This has the {\it same} form as the Newtonian approximation -- but the latter is arrived at by implicitly neglecting the contribution of  the metric perturbation $\mathcal{B}$.}

At linear order, the GR and Newtonian results agree: $\delta^{\text{\tiny ({{1}})}} _{\rm{mC}} =\delta^{\text{\tiny ({{1}})}} _{\rm{mN,C}} = \delta_{\rm{m}}^{\text{\tiny ({{1}})}}$.
{At second order, we can decompose $\delta_{\rm{mC}}$ into a Newtonian part and a GR correction $\delta^{\text{\tiny ({{2}})}}_{\rm{mC}}= \delta_{\rm{mN,C}}^{\text{\tiny ({{2}})}} +\delta_{{\rm{mGR,C}}}^{\text{\tiny ({{2}})}}$}, where the Newtonian part is given by
\begin{eqnarray}
 {\delta_{\rm{mN,C}}^{\text{\tiny ({{2}})}}}({\tau},\bm{q})&=&
 \frac{2}{3} \left[2+\frac{{ F{(\tau)}}}{{ D{(\tau)}}^2}\right] \big(\delta^{\text{\tiny ({{1}})}}_{\rm{m}}({\tau},\bm{q})\big)^2  + \left[1 - \frac{{F{(\tau)}}}{{D{(\tau)}}^2}\right] s^2({\tau},\bm{q})\,.
    \label{eq:secondorderdensityCSgauge}
\end{eqnarray}
Here  $D$ is the matter growth factor at linear order and $s^2 = s_{ij}s^{ij}$, where the tidal field $s_{ij}$ is defined below.   In an Einstein-de Sitter background, ${F}(\tau)=3[{D}(\tau)]^2/7$, and this is a very good approximation in $\Lambda$CDM as well.  The second-order GR correction
 is given by \cite{Bruni:2013qta,Villa:2015ppa}
\begin{eqnarray}
\delta^{\text{\tiny ({{2}})}}_{{\rm{{mGR,C}}}}({\bm{q}})={ 6\Omega_{\rm m } \mathcal{H}^2} \left[ -\frac{1}{4}\partial_i \nabla^{-2} \delta^{\text{\tiny ({{1}})}}_{\rm{m}}({\bm{q}}) \partial^i \nabla^{-2} \delta^{\text{\tiny ({{1}})}}_{\rm{m}}({\bm{q}})
    +\delta^{\text{\tiny ({{1}})}}_{\rm{m}} ({\bm{q}})\nabla^{-2} \delta^{\text{\tiny ({{1}})}}_{\rm{m}}({\bm{q}}) \right] \Big(1+\frac{2}{3}{f\over\Omega_{\rm{m}}}\Big)\,,
    \label{eq:GRdensity}
\end{eqnarray}
where  $\Omega_{\rm{m}}$ is the matter-energy density parameter,  $f$ is the growth rate; in a $\Lambda$CDM universe $f = \Omega_{\rm{m}}^{0.55}$ is a good approximation.   {{We have introduced the conformal Hubble parameter $\mathcal{H} = \mathcal{H}(\eta)$, where $\eta$ is the conformal time,  related to $\tau$ according to $d\tau = a d \eta$}}.  $\nabla^{-2} \delta^{\text{\tiny ({{1}})}}_{\rm{m}}({\bm{q}})  \sim \Phi({\bm{q}})$ and $\partial_i \nabla^{-2} \delta^{\text{\tiny ({{1}})}}_{\rm{m}}({\bm{q}}) \sim \partial_i v({\bm{q}})$, {with $v$ being  the peculiar velocity potential}. {We have suppressed the $\tau$ dependence for brevity.}

\subsection{ Effective theory-like description of galaxy formation}
In the previous section, we provided the relation between the initial and evolved galaxy density in C-gauge. In this section, we  provide the expression for the initial galaxy density. The idea here is to describe the complicated physics of galaxy formation using a set of bias parameters that relates the galaxy density to a set of local observables constructed from the initial curvature perturbation. The bias parameters hide our ignorance of the detailed physics of galaxy formation. The key difficulty though is how to define these local observables, since we cannot simply expand $\delta_{\rm{g}}^{\rm{L}}$ as a functional of a set of all terms that could be formed from $\Phi$, for example  $ \mathcal{F}(\Phi,\partial _i\Phi, \partial_{i}\partial_{j}\Phi)$.  The reason is that the gradient of $\Phi$ is coordinate dependent and the relationship between $\Phi$ and $\delta_{\rm{mC}}$ in GR differs from its equivalent in the Newtonian limit. We first review the Newtonian case.

\begin{figure}[!h]

\includegraphics[width=140mm] {longshort2.png}
\caption{The thick grey line is the long mode of the dark matter density field, while the snaky black line is the total dark matter density fluctuation. The short modes are responsible for the intrinsic scatter around $\delta_{\l}$.
 }
\label{fig:patches}
\end{figure}

\begin{itemize}

\item {\bf{Lagrangian bias in the Newtonian approximation:}}
The initial galaxy density may be expressed in terms of the  local observables  constructed from the initial metric fluctuation, $\delta_{\rm{g}}^{\rm{L}}({\bm{q}}) =\mathcal{F}[\Phi^{\rm{ini}}({\bm{q}})]$, but we need to remove unphysical modes. We make a local gradient expansion of $\Phi^{\rm{ini}}({\bm{q}})$ about a reference world-line $\bm{q} = 0$,
\begin{eqnarray}
 {\Phi^{\rm{ini}}}({\bm{q}}) &=& \Phi^{\rm{ini}}_0+( \partial_{i}{\Phi^{\rm{ini}}_0}) q^{i}
  + \frac{1}{2}(\partial_{i}\partial_{j}{\Phi^{\rm{ini}}_0}) q^{i}q^{j}  + \mathcal{O}({\bm{q}})^{3} \,,
  \label{eq:Taylorseries}
\end{eqnarray}
where we set $\Phi_{0}^{\rm{ini}} \equiv \Phi^{\rm{ini}}(\bm{0})$, $ \partial_{i}{\Phi^{\rm{ini}}_0} \equiv\left( \partial_{i}{\Phi^{\rm{ini}}}\right)_0$  and $\partial_{i}\partial_{j}{\Phi^{\rm{ini}}_0} \equiv \left(\partial_{i}\partial_{j}{\Phi^{\rm{ini}}}\right)_0$ to reduce clutter.
The second term does not have any influence on physics within the local patch as it can be removed by a change of coordinates,
\begin{eqnarray} \label{eq:coordaintetransform}
\tilde{\eta} = \eta\,,\qquad \qquad
\tilde{q} ^i= {q}^i + {{\xi}}^i(\eta)\,,
\end{eqnarray}
where $\xi$ is a spatially constant 3-vector. Under this coordinate change, $\Phi$ transforms as \cite{Kehagias:2013yd}
\begin{eqnarray}
{\Phi^{\rm{ini}}}(\tilde{\bm{q}}) = \Phi^{\rm{ini}}({\bm{q}}) - \left( \partial_{\eta}^{2} {\xi}_{i}+ \mathcal{H} \partial_{\eta} {\xi}_{{i}}\right)  {q}^i.
\label{eq:NewtonianConfKillingeqn}
\end{eqnarray}
The constant gradient $\partial_{i}\Phi^{\rm{ini}}_0$ can be removed by demanding that  ${{\xi}^i}$ solves the equation
\begin{equation}
\partial_{\eta}^{2} {\xi}^{i}+ \mathcal{H} \partial_{\eta} {\xi}^{{i}}= \partial^{i}\Phi^{\rm{ini}}_0.
\end{equation}
In the new coordinates, the gradient term vanishes and the first term is simply a constant and may be absorbed into the scale factor of the background spacetime.
The coordinate transformation that removes this term corresponds to  {{re-labelling of the fluid world-lines}} in the presence of the long-wavelength mode~\cite{Desjacques:2016bnm}. In the new coordinates, the galaxy position is slightly displaced, ${\bm{q}} + {\bf{\xi}} = \tilde{\bm{q}}\rightarrow {\bf{\xi}}$,  and this coordinate change leaves equation \eqref{eq:conservationgen}  invariant.
Thus, $\delta_{\rm{g}}^{\rm{L}} $ is only a functional of $\partial_{i}\partial_{j}\Phi^{\rm{ini}}_0 $ and its higher derivatives.  The observables are then constructed from the irreducible  decomposition
\begin{eqnarray}
\partial_{i}\partial_{j}\Phi^{\rm{ini}}_0  = \frac{1}{3}\nabla^2 \Phi_0^{\rm{ini}}\delta_{ij} + s_{0ij} \,,
\label{eq:irrducibledecomp}
\end{eqnarray}
where $s_{0ij}$ is the initial tidal tensor: $s_{0ij} = {D}_{ij} \Phi^{\rm{ini}}_0$ and ${D}_{ij} $ is given by ${D}_{ij} =  {\partial_{i}\partial_{j}} - {\nabla^{2}} \delta_{ij}/3$ .  Furthermore, the energy density constraint equation  in the Newtonian approximation is the Poisson equation
\begin{eqnarray}
 {\nabla^2 \Phi} (\tilde{\bm{q}}) = \frac{3}{2} {{\Omega_{\rm{m}} \mathcal{H}^2}}\delta_{{\rm{mN}}}(\tilde{\bm{q}}) \,, \label{eq:firstorderdensityCSgauge}
\end{eqnarray}
which gives a linear relationship between $\Phi$ and $\delta_{\rm{mN}}$ at all orders in perturbation theory~\cite{Bernardeau:2001qr}.
By decomposing $\delta_{{\rm{mN}}}$ into long and short modes, $\delta_{{\rm{mN}}}  = \delta_s + \delta_{\l}$, it is possible to  specify $\delta_{\rm{g}}^{\rm{L}}(\tilde{\bm{q}})$ as a local functional of the long mode, $
\delta_{\rm{g}}^{\rm{L
}}(\tilde{\bm{q}}) = \mathcal{F}\left[\delta_{\l}(\tilde{\bm{q}}), s^2_{\l}(\tilde{\bm{q}})\right] \,,
$
where the effect of the short mode is encoded in the bias parameters.

\item {\bf{Lagrangian bias in general relativity}:}
We now turn into the case of GR. In GR, the energy constraint equation that links $\Phi$ to $\delta_{\rm{ m}}$ is a nonlinear equation\footnote{This is derived from the Hamiltonian constraint in a $1+3$ covariant decomposition~\cite{Ellis1971grc..conf..104E,Ellis:1998ct}.}. At second order in C-gauge it is given by~\cite{Bruni:2014xma,Bartolo:2015qva}
\begin{eqnarray}
\nabla^2 \Phi ({\bm{q}}) +\bigg[2 \Phi ({\bm{q}})\nabla^2 \Phi({\bm{q}}) - \frac{1}{2}\partial_i\Phi ({\bm{q}})\partial^i \Phi({\bm{q}})\bigg]\left(1 + \frac{2}{3} \frac{f}{\Omega_{\rm{m}}}\right) = \frac{3}{2} \Omega_{\rm{m}} \mathcal{H}^2 \delta_{\rm{mC}}({\bm{q}})\,.
\label{eq:GRPoissoneqn}
\end{eqnarray}
An equivalent expression exists in Possion gauge~\cite{Hidalgo:2013mba} and in total matter gauge~\cite{Villa:2015ppa}.
We perform a short/long-wavelength mode decomposition,
neglecting gradients of the long modes relative to those of the short modes, and using $\Phi_s\nabla^2\Phi_{l} \ll \Phi_l\nabla^2 \Phi_s$. This leads to
\begin{eqnarray}
\nabla^2 \Phi_s({\bm{q}})  +\bigg[ 2 \Phi_{\l}({\bm{q}}) \nabla^2 \Phi_s ({\bm{q}})- \partial_i\Phi_{\l}({\bm{q}}) \partial^i \Phi_s({\bm{q}}) \bigg]\left(1 + \frac{2}{3} \frac{f}{\Omega_{\rm{m}}}\right)= \frac{3}{2} \Omega_{\rm{m}} \mathcal{H}^2 \delta_{\rm{mC}s}({\bm{q}})\,,
\label{eq:longshortcoupling}
\end{eqnarray}
for the short mode.
This equation may suggest that clustering on small scales is modulated by the long-wavelength mode in a similar way to primordial non-Gaussianity of the local type~\cite{Bruni:2014xma}, and this would in principle require additional bias parameters to capture these effects.

However, this is not the case as we now show.  Firstly, we have to elevate the  coordinate transformation (equation \eqref{eq:coordaintetransform}) that removes the unphysical mode in the Newtonian limit to full GR. Similar to the Newtonian approach, we consider the following coordinate transformation
\begin{eqnarray}
\tilde{q} ^{i}&= &{q}^{i} + {{\xi}}^{i}.
\label{eq:localtrajectory}
\end{eqnarray}
Under this change of coordinates the spatial metric transforms as
\begin{eqnarray}
\tilde{\gamma}_{ij} - {\gamma}_{ij} =  \xi^{k} \partial_{k} {\gamma}_{ij} + {\gamma}_{kj} \partial_{i} \xi^{k} + {\gamma}_{ik} \partial_{j} \xi^{k} \,,
\label{eq:metrictransform}
\end{eqnarray}
where the right-hand side is equal to the Lie derivative of $\gamma_{ij}$, $\mathcal{L}_{\xi} \gamma_{ij}$.
We then seek  $\xi^{i}$ that solves the following conformal Killing equation
\begin{eqnarray}
 \mathcal{L}_{\xi} \gamma_{ij} = \left( \mathcal{A} -1\right)\gamma_{ij} \,,

 \label{eq:CKE}
\end{eqnarray}
such that the metric transforms as $\tilde{\gamma}_{ij} = \mathcal{A}{\gamma}_{ij}$, where $\mathcal{A}$ is an effective conformal factor. $\xi$ that solves equation \eqref{eq:CKE} for the Euclidean metric, $\delta_{ij}$, is given by \cite{levine1936,levine1939}
\begin{eqnarray}
{\xi}^{i} &=&  a^{i} + M^{i}{}_{j} q^{j} + \lambda q^{i}+ 2 \left( q_{j} b^{j}\right) q^{i} -  q_jq^j b^{i}\,,
\label{eq:solntoCKE}
\end{eqnarray}
where $a^{i}$, $b^{j}$ and $\lambda$ are infinitesimally small and spatially constant.  $M^{i}{}_{j} $ and $a^{i} $ are associated with the spatial rotations and translations respectively, $\lambda$ is related to the dilatation   and $b^{j}$ is related to the  special conformal transformation.  The effective conformal factor may then be expressed in terms of $\xi^{i}$ as
\begin{eqnarray}
\mathcal{A}= 1  +  \frac{2}{3}\partial^{i} \xi_{i} = 1+2\left(\lambda + 2 q^{i} b_{i}\right)\,.
\label{A}
\end{eqnarray}
Dropping the spatial rotations and translations since they leave the metric unchanged,  equation \eqref{eq:localtrajectory} becomes
\begin{eqnarray} \label{eq:etacoord2}
\tilde{\eta}&=&\eta \,,
\\
 \tilde{q}^i&=&{q}^i + \lambda q^i + 2 q^i  q^jb_j -{q}_j {q}^j b^i\,.
\label{eq:xcoord2}
\end{eqnarray}

Now applying the spatial diffeomorphism to the initial metric perturbation,
 \begin{eqnarray}
 \gamma_{ij}^{\rm{ini}} =\delta_{ij} {\exp}\left(-\frac{10}{3} \Phi^{\rm{ini}}({{\bm{q}}})\right)\,,
 \end{eqnarray}
 and using equation (\ref{A}), we can determine $\lambda$ and $b^{j}$ that remove the conformal factor of the initial metric for constant $\Phi^{\rm{ini}}_{0}$ and $\partial_i\Phi^{\rm{ini}}_{0}$. This gives the transformation
\begin{eqnarray}
\tilde{\eta}& =& \eta \,,
\\
\tilde{q}^i &=&q^i (1- \frac{5}{3}\Phi^{\rm{ini}}_{0})  -\frac{5}{3} q^i {q}^j \partial_j \Phi^{\rm{ini}}_{0}  + \frac{5}{6} q_j q^j\partial^i \Phi^{\rm{ini}}_{0}\,.
\label{eq:xcoordbody}
\end{eqnarray}
$\Phi^{\rm{ini}}_{0}$ is related to the long-wavelength mode of the matter density perturbation \cite{Tram:2016cpy,Koyama:2018ttg}
\begin{eqnarray}
\Phi^{\rm{ini}}_{0}=\frac{9}{10} \Omega_{\rm{m}} \mathcal{H}^2 \left[1+ \frac{2f}{3 \Omega_{\rm{m}}} \right] \nabla^{-2}\delta_0\,,
\label{eq:zetatodelta_L}
\end{eqnarray}
where $\delta_0$ is the leading order term in the gradient expansion of $\delta_{\l}$
\begin{equation}
\delta_l({\bm{q}}) = \delta_0+ (\partial_{i}{\delta_0}) q^{i} + \cdots.
\end{equation}
Again we set $\delta_0\equiv \delta({\bf{0)}}$ and  $\partial_{i}{\delta_0 }\equiv (\partial_{i}{\delta})_0$.
{In Appendix~\ref{sec:CFC}, we show that this coordinate transformation is equivalent at this order to going to the conformal Fermi coordinates defined by
  \cite{Pajer:2013ana,Dai:2015rda,Dai:2015jaa}.}

Under the local coordinate transformation given in equation \eqref{eq:xcoordbody}, the linear matter density transforms as
\begin{eqnarray}
 \delta_{\rm{mC}}(  \tilde{\bm{q}})  = \delta_{\rm{mC}} ({\bm{q}}) + \partial_j \delta_{\rm{mC}}\left({\bm{q}}\right)  \left(  \tilde{q}^j-{q}^j\right)\,.
\label{eq:CSgaugemap}
\end{eqnarray}
We split the density into short and long modes as $\delta_{\rm{mC}} = \delta_{l} + \delta_s$ and perform the transformation for the short mode density.
After lengthy algebra (see details in {{Appendix \ref{sec:generallongmode}}}), equation \eqref{eq:CSgaugemap} becomes
\begin{eqnarray}
\delta_s( \tilde{\bm{q}}) & =&  \delta_s({\bm{q}})
 +{3}{\Omega_{\rm{m}} \mathcal{H}^2}\left[ 1 + \frac{2 f}{3 \Omega_{\rm{m}}}   \right]\delta_{{s}}({\bm{q}}) \nabla^{-2} \delta_0
+\frac{3}{2} \Omega_{\rm{m}} \mathcal{H}^2\left[1+ \frac{2f}{3 \Omega_{\rm{m}}} \right] \partial^j\nabla^{-2} \delta_{s}({\bm{q}})\partial_j  \nabla^{-2}\delta_{\rm{0}}\,.
\label{eq:densitycoord}
\end{eqnarray}
Substituting equation \eqref{eq:densitycoord} in equation \eqref{eq:longshortcoupling}  removes all the long-short coupling contribution therein. Thus, in local coordinates or within the local patch, the matter density contrast at second order in C-gauge is well approximated by the Newtonian density field and the local patch is not modulated by the long mode, hence  no new bias parameter  is required. We reach the same conclusion by constructing local coordinates using conformal Fermi coordinates (CFC) as shown in Appendix~\ref{sec:CFC}.

\end{itemize}

We can now specify  $\delta^{\text{L}}_{\rm{g}}({\tilde{\bm{q}}}) $ as a functional of local observables constructed from the second derivatives of $\Phi_l$:
\begin{eqnarray}
1+\delta^{\text{L}}_{\rm{g}}(\tilde{\bm{q}}) = \mathcal{F}\left[\delta_{\l}(\tilde{\bm{q}}), s^2_{\l}(\tilde{\bm{q}})\right] = 1+b^\text{L}_{1}\delta_{\l}^{\text{\tiny ({{1}})}}(\tilde{\bm{q}})
+\frac{1}{2}\left[b_{2}^\text{L}(\delta_{\l}^{\text{\tiny ({{1}})}}(\tilde{\bm{q}}))^2+ b_{s}^\text{L}s^2_{\l}(\tilde{\bm{q}})\right]\,,
\label{eq:halooverdensity}
\end{eqnarray}
where $b^\text{L}_{1}$  and $b_{2}^\text{L}$ are linear and nonlinear Lagrangian bias parameters respectively and $b_{s}^\text{L}$ is the initial tidal bias parameter~\cite{Desjacques:2016bnm};
\begin{eqnarray}
b^\text{L}_{1} \equiv\frac{D_{\rm{ini}}}{D} \frac{\partial {\mathcal{F}}}{\partial \delta_{\l}}
\,, \qquad
b^\text{L}_{2} \equiv\left(\frac{D_{\rm{ini}}}{D}\right)^2 \frac{\partial^2 {\mathcal{F}}}{\partial \delta_{\l}^2}
\,, \qquad
b^\text{L}_{s} \equiv\left(\frac{D_{\rm{ini}}}{D}\right)^2 \frac{\partial {\mathcal{F}}}{\partial s^2_{\l}}\,.
\end{eqnarray}
The evolution of the  galaxy number is dependent on the hypersurface or the trajectory of the fluid flow, so we need to transform equation \eqref{eq:conservationgen} to the local coordinates using  equation \eqref{eq:localtrajectory};
\begin{eqnarray}
1+ \delta_{{\rm{gC}}\l}( \tilde{\bm{q}})& =&
 \left[ 1 + \delta^{\text{L}}_{\rm{g}}(\tilde{\bm{q}})\right]\left[1 + \delta_{\rm{mC}}(\tau, \tilde{\bm{q}}) \right]
\\
& =&1 + \left[1 + b_1^{\rm{L}}\right]\delta_{\l}^{\text{\tiny ({{1}})}}(\tilde{\bm{q}})
+ \frac{1}{2} \left[\delta^{\text{\tiny ({{2}})}}_{{\rm mN} \l} (\tilde{\bm{q}}) +\delta^{\text{\tiny ({{2}})}}_{{\rm{mGR,C}} \l}(\tilde{\bm{q}})+  \left[b_2^{\rm{L}} +2 b_1^{\rm{L}}\right]\left(\delta_{\l}^{\text{\tiny ({{1}})}}(\tilde{\bm{q}})\right)^2 +  b_{s}^{\rm{L}}s^2_{\l}(\tilde{\bm{q}})\right]\,.
\label{eq:galaxibias}
\end{eqnarray}
In the first line, we used the fact that the conservation equation is unchanged under the local coordinate transformation. In the second line, we made use of equation \eqref{eq:halooverdensity} and the long-wavelength part of $\delta_{\rm{mC}}$.
 To obtain the long-wavelength part of $\delta_{\rm{mC}}$ at second order, i.e $\delta^{\text{\tiny ({{2}})}}_{{\rm mN} \l} $ and $\delta^{\text{\tiny ({{2}})}}_{{\rm{mGR,C}} \l}$,  we first decompose equation \eqref{eq:GRdensity} into long and short-wavelength parts and transform to local coordinates. Long/short-wavelength coupling  terms in $\delta^{\text{\tiny ({{2}})}}_{{\rm mN}} $ are absorbed into the local spatial curvature \cite{Dai:2015jaa}, {{the short/short-wavelength coupling terms in $\delta^{\text{\tiny ({{2}})}}_{{\rm{mGR,C}}}$ are negligible as $\nabla^{-2} \delta^{\text{\tiny ({{1}})}}_{\l} \gg \nabla^{-2} \delta^{\text{\tiny ({{1}})}}_{s}$}}, while the long/short-wavelength coupling  terms in $\delta^{\text{\tiny ({{2}})}}_{{\rm{mGR,C}}}$  are removed by local  coordinate transformation~(see equation \eqref{eq:densitycoord}),  leaving
\begin{eqnarray}    \label{eq:longlongGRdensity}
\delta_{{\rm{mGR,C}} \l} ^{\text{\tiny ({{2}})}}(\tilde{\bm{q}})=6{ \Omega_{\rm m}\mathcal{H}^2} \left[ -\frac{1}{4}\partial_i \nabla^{-2} \delta^{\text{\tiny ({{1}})}}_{\l}(\tilde{\bm{q}}) \partial^i \nabla^{-2} \delta^{\text{\tiny ({{1}})}}_{\l}(\tilde{\bm{q}})
    +\delta^{\text{\tiny ({{1}})}}_{\l} (\tilde{\bm{q}})\nabla^{-2} \delta^{\text{\tiny ({{1}})}}_{\l}(\tilde{\bm{q}}) \right] \Big(1+\frac{2}{3}{f\over\Omega_{\rm{m}}}\Big)\,.
\end{eqnarray}
This is how equation \eqref{eq:galaxibias} receives a general relativistic correction. Note that this contribution is not coming from the initial galaxy density, rather it is coming from the effect of volume distortions as the tracer evolves from $\tau_{\rm{ini}} $ to $\tau$.
We  may now simplify equation \eqref{eq:galaxibias} further such that the second-order matter density contrast appears in the same form as the linear order term, i.e $\left[1 + b_1^{\rm{L}}\right]\delta_{\l}^{\text{\tiny ({{1}})}}(\tilde{\bm{q}})$. A few steps of algebraic simplification lead to
\begin{eqnarray} \label{eq:galaxyconservation2}
1+ \delta_{\!{\rm{gC}}\l}(\tilde{\bm{q}})& =&1 + \left[1 + b_1^{\rm{L}}\right]\delta_{\l}^{\text{\tiny ({{1}})}} (\tilde{\bm{q}})
\\ \nonumber &&
+ \frac{1}{2} \bigg\{\left(1 + b_1^{\rm{L}}\right) \delta^{\text{\tiny ({{2}})}}_{{\rm{m N}} \l}  (\tilde{\bm{q}})
 + \left[ b_{2}^{\text{L}} + \frac{2}{3}b^{\text{L}} _1\left(1-\frac{{F}}{{D}^2}\right) \right]\left(\delta^{\text{\tiny ({{1}})}}_{\l}(\tilde{\bm{q}})\right)^2  + \left[  b_{s}^{\text{L}} -b_{1}^\text{L} \left(1-\frac{{F}}{{D}^2}\right)\right] s^2_{\l}(\bm{q})+\delta_{{\rm{mGR,C}} \l}^{\text{\tiny ({{2}})}}(\tilde{\bm{q}})\bigg\}\,.
\end{eqnarray}
We can now change $\tilde{q}$ to $q$ via coordinate transformation;
for $\delta_{{\rm{gC} }\l }$ we have
\begin{eqnarray}
\delta_{{\rm{gC}} \l }( {\bm{q}})  &=& \delta_{{\rm{gC}} \l} (\tilde{\bm{q}}) - \partial_j \delta_{{\rm{gC}}\l}\left({\bm{q}}\right)  \left(  \tilde{q}^j-{q}^j\right)\,.
\end{eqnarray}
There is a similar expression for $\delta_{\l}( {\bm{q}}) $.  Applying these to equation \eqref{eq:galaxyconservation2} and requiring that equation \eqref{eq:galaxyconservation2} is satisfied order by order, i.e $ \partial_j \delta_{{\rm{gC}} \l }^{\text{\tiny ({{1}})}} =\left[1 + b_{1}^{\rm{L}}\right](\partial_{i}{\delta_{\l}^{\text{\tiny ({{1}})}}})$, leads to
\begin{eqnarray} \label{eq:localbias}
\delta_{\!\rm{gC}}({\bm{q}})&=&\left[1 + b_1^{\rm{L}}\right]\left[ \delta_{\rm{m N}}^{\text{\tiny ({{1}})}}(\bm{q}) + \frac{1}{2}\delta_{\rm{m N}}^{\text{\tiny ({{2}})}}(\bm{q}) \right]
\\ \nonumber &&
 +\frac{1}{2}\left[ \left[ b_{2}^{\text{L}} + \frac{2}{3}b^{\text{L}} _1\left(1-\frac{{F}}{{D}^2}\right) \right]\left(\delta_{\rm{m}}^{\text{\tiny ({{1}})}}(\bm{q})\right)^2  +  \left[  b_{s}^{\text{L}} -b_{1}^\text{L} \left(1-\frac{{F}}{{D}^2}\right)\right] s^2(\bm{q}) +\delta_{{\rm{mGR,C}}}^{\text{\tiny ({{2}})}}(\bm{q}) \right]\,.
\end{eqnarray}
We  omit the subscript $\l$ from now on as all perturbations in the above equation are long-mode and there is no confusion.

  \section{Eulerian gauges in global coordinates}
We have specified the galaxy density contrast in terms of the long-wavelength mode of the dark matter density contrast in C-gauge. As mentioned in  the introduction, in GR, there is no unique Eulerian frame. We can go to any convenient gauge by a performing coordinate transformation. In GR, perturbations change under a general coordinate transformation
\begin{eqnarray}
 {x}^{\mu} \rightarrow x^{\mu} + Z^{\mu}\,, \qquad {\rm{with}} \qquad  Z^{\mu} = ( T,L^i).
 \label{eq:gaugetransform}
\end{eqnarray}
Here $T$ stands for a temporal gauge choice and $L^i$ corresponds to spatial gauge choice and we decompose $L^i =\partial^i L+ L_{\bot}^i $, where $\partial _i L_{\bot}^i =0$.
Consider a perturbed line element on an FLRW background
\begin{eqnarray}
ds^2 = a^2 \Big[   - (1 + 2 \psi) d \eta^2 + 2 B_i dx^i d \eta + (1 - 2 \phi) \delta_{ij} dx^i dx^j
+ 2 E_{ij} dx^i dx^j \Big]\,,
\end{eqnarray}
 {where $B_i=\partial_iB$ and $E_{ij} =\big(\partial_i \partial_j - \frac{1}{3} \delta_{ij} \nabla^2 \big) E.$}
 We consider two different  gauge choices, $X$-gauge and C-gauge. The gauge transformation  of the  density field from $X$ to C-gauge up to second order is given by  \cite{Bruni:1996im,Villa:2015ppa}
 \begin{eqnarray}\label{eq:Xgauge}
{\delta}_{ \rm{IX}}({\bm{x}})&=&\delta_{\rm{ IC}}({\bm{x}})- 3 {\cal H}T({\bm{x}})+\frac{1}{2}T({\bm{x}})\bigg[ 3\left( 2 {\cal H}^2 + \frac{3}{ 2} {\mathcal H^{2}\Omega_{{\rm m}}} \right)\,T({\bm{x}}) - 3 {\cal H}{T'({\bm{x}})}
\\ \nonumber &&
+2\delta_{\rm{IC}}'({\bm{x}}) - 6 {\cal H} \delta_{\rm{IC}}({\bm{x}}) \bigg]
+\left(\partial_i\delta_{ \rm{IC}}({\bm{x}}) - \frac{3}{2}{\cal H}\partial_i T({\bm{x}})\right)\partial^iL({\bm{x}})\,,
 \end{eqnarray}
where ${\rm{I}} = {\rm{g, m}}$ and $' = \partial_{\eta}$ and we omit the  $\eta$ dependence. Note that both ${\delta}_{ \rm{IC}}$ and $\delta_{ \rm{IX}}$ are calculated at the same coordinate position.
We consider the total matter gauge (T-gauge) which corresponds to an Eulerian gauge in the  Newtonian approximation and then give a general expression for any other Eulerian gauge choice.

\begin{itemize}
\item {\tt{Total matter gauge:}} In T-gauge, the components of the gauge vector are \cite{Villa:2015ppa}
\begin{eqnarray}\label{eq:gaugegenCT1}
L^{\text{\tiny ({{1}})}}({\bm{x}})&=& \ \nabla^{-2}\delta_{\rm{m}}^{\text{\tiny ({{1}})}}({\bm{x}}) \,, \qquad \qquad
T = 0\,.\label{eq:gaugegenCT1B}
\end{eqnarray}
At  linear order the density does not change and at the second order,
\begin{eqnarray}
\delta^{\text{\tiny ({{2}})}}_{\rm{gT}}({\bm{x}}) &= &  \delta^{\text{\tiny ({{2}})}}_{\rm{gC}}({\bm{q}})+2{\partial^j\nabla^{-2} \delta^{\text{\tiny ({{1}})}}_{\rm{m}}}({\bm{x}}) \partial_j \delta^{\text{\tiny ({{1}})}}_{\rm{gC}}({\bm{x}})\,,
\end{eqnarray}
 Putting everything together, the galaxy bias model in T-gauge becomes
\begin{eqnarray}\label{eq:Eulerianbias}
\delta_{\rm{gT}}({\bm{x}}) &=&
b_{1}\left[\delta^{\text{\tiny ({{1}})}}_{\rm{m}}({\bm{x}}) + \frac{1}{2}\delta^{\text{\tiny ({{2}})}}_{\rm{m N,T}}({\bm{x}})\right]
+\frac{1}{2}\bigg[b_{2}[\delta_{m}^{\text{\tiny ({{1}})}}({\bm{x}})]^2
 +b_{s} s^2({\bm{x}}) +{\delta_{\rm{mGR,C}}^{\text{\tiny ({{2}})}}({\bm{x}})} \bigg]\,,
\end{eqnarray}
 where we introduced the `Eulerian'  bias parameters
\begin{eqnarray}
b_{1}&=&1+b_{1}^\text{L}\,,\\
b_{2}&=&b_{2}^{\text{L}} + \frac{2}{3}\left(b_{1} -1\right)\left[1-\frac{{F}}{{D}^2}\right]=\frac{8}{21}\left(b_{1}-1\right)+b_{2}^\text{L}\,,\\
b_{s} &=& b_{s}^{\text{L}} -\left(b_{1} -1\right)\left[1-\frac{{F}}{{D}^2}\right]=b_{s}^\text{L}-\frac{4}{7}\left(b_{1} -1\right)\,.
\end{eqnarray}
The second equality holds in the Einstein de Sitter limit.
$\delta^{\text{\tiny ({{2}})}}_{\rm{ mN,T}}$ is the Newtonian limit of the second order matter density perturbation in T-gauge,
\begin{eqnarray}
\delta^{\text{\tiny ({{2}})}}_{\rm{ mN,T}}({\bm{x}})&= &
2\partial_j \nabla^{-2}\delta_{\rm{m}}^{\text{\tiny ({{1}})}}({\bm{x}}) \partial_j \delta_{\rm{m}}^{\text{\tiny ({{1}})}}({\bm{x}})+ \frac{2}{3} \left(2+ \frac{F}{D^2}\right)  (\delta_{\rm{m}} ^{\text{\tiny ({{1}})}}({\bm{x}}))^2+  \left(1-\frac{F}{D^2}\right) s^2({\bm{x}})  \,.
\label{eq:TgaugeNewtonian}
\end{eqnarray}
This agrees with the conventional Newtonian result in the Eulerian frame.

\item {\tt{X-Eulerian density:}}

Using  equation \eqref{eq:localbias} in equation \eqref{eq:Xgauge} gives
the general Eulerian galaxy density in $X-$gauge

\begin{eqnarray}\label{eq:GaldensityinX}
{\delta}_{\rm{gX}}({\bm{x}})&=&b_1\left[\delta^{\text{\tiny ({{1}})}}_{\rm{mX}}({\bm{x}})+\frac{1}{2}\delta^{\text{\tiny ({{2}})}}_{\rm{mX}}({\bm{x}})\right]+ \left(b_1-1\right) \left[{3}\mathcal{H} T^{\text{\tiny ({{1}})}}({\bm{x}})-\frac{1}{2}\left(\delta_{\rm{mGR,C}}^{\text{\tiny ({{2}})}}({\bm{x}})- {3}\mathcal{H} T^{\text{\tiny ({{2}})}}({\bm{x}})\right)\right]
\\ \nonumber &&
+\frac{1}{2}\left[ b_2\left(\delta^{\text{\tiny ({{1}})}}_{\rm{mX}}({\bm{x}})\right)^2  + b_{s}s^2({\bm{x}})\right] + \left( b_1' + 3 \mathcal{H} b_2\right) \delta^{\text{\tiny ({{1}})}}_{\rm{mX}}({\bm{x}})T^{\text{\tiny ({{1}})}}({\bm{x}})
\\ \nonumber &&
+\frac{3}{2}\left(b_1 -1\right) \bigg[ \mathcal{H} \partial_iT^{\text{\tiny ({{1}})}}({\bm{x}}) \partial^i L^{\text{\tiny ({{1}})}} ({\bm{x}})
- {T^{\text{\tiny ({{1}})}}}'({\bm{x}})T^{\text{\tiny ({{1}})}}({\bm{x}}) \bigg]
+\frac{3}{2}\bigg[\left(b_1-1\right) \left( 2 {\cal H}^2
+ \frac{3}{ 2} {\mathcal H^{2}\Omega_{{\rm m}}} \right)\,
\\ \nonumber &&
 + b_1' \mathcal{H} + 3b_2 \mathcal{H}^2\bigg](T^{\text{\tiny ({{1}})}}({\bm{x}}))^2
\,.
\end{eqnarray}
Note that when $\delta^{\text{\tiny ({{2}})}}_{\rm{mX}}({\bm{x}})$ is decomposed into Newtonian and GR corrections parts, $b_1\delta_{\rm{mGR,C}}^{\text{\tiny ({{2}})}}({\bm{x}})$ drops out.
The possible gauge choices include N-body-gauge for interpreting the Newtonian N-body simulation in the general relativistic context \cite{Fidler:2017pnb}  and  N-Boisson-gauge, which allows to  include the effects of radiation in the Newtonian N-body simulation \cite{Fidler:2018geb}.
\end{itemize}

{{Equation \eqref{eq:localbias}, with equation \eqref{eq:longlongGRdensity}, is our key result.}}

\section{Discussion and Conclusion}

It is well-known in cosmological perturbation theory that the comoving-synchronous  gauge completely fixes the gauge in a matter  {{plus cosmological constant}} dominated universe, {{once the initial matter world-line coordinates are fixed}}~\cite{Hwang:1999yv} .  So where does the extra freedom to re-define the coordinates  come from? It is important to stress that there is no contradiction with standard cosmological perturbation theory, since it assumes that the initial curvature perturbation ($\zeta = - 5\Phi^{\rm{ini}}/3$) falls off appropriately as $r \rightarrow \infty$~\cite{Hinterbichler:2012nm}. In our case, we made a key assumption that the galaxy formation process happens within a local patch of the  full spacetime. Based on this assumption, we split the initial curvature perturbation into long and short-wavelength modes, $\zeta = \zeta_s + \zeta_{\l}$, where short-wavelength refers to the modes that are equal or shorter than the size of the local patch. This allows us to express the initial metric  as
\begin{eqnarray}
\gamma_{ij}^{\rm{ini}} = \delta_{ij} e^{2\zeta }= \delta_{ij} e^{2\zeta_{s}}e^{2\zeta_{\l}}.
\end{eqnarray}
The exponential conformal factor for the short-wavelength mode falls off in the limit $r\rightarrow R$.
The exponential conformal factor for the long-wavelength mode does not die off in this limit, hence it is unobservable within the local patch.

 As a result it generates conformal transformations of the local patch which results in a residual diffeomorphism~\cite{Hinterbichler:2012nm,Weinberg:2003sw,Creminelli:2013mca}. We have  used this residual diffeomorphism symmetry to absorb $\zeta_{\l}$ into the coordinates of the FLRW background spacetime. If both $ \zeta_s$  and $\zeta_{\l}$ fall off as $r\rightarrow R$, there will not be any residual symmetry. Only in this limit can the comoving-synchronous gauge completely fix the gauge in the matter plus cosmological constant dominated universe.
We used these residual diffeomorphism symmetries of GR to show that there is no coupling between $ \zeta_s$  and $\zeta_{\l}$ in the energy constraint equation in GR for a $\Lambda$CDM universe. This implies that there is no modulation of the local physics of galaxy formation by the long-wavelength mode of the metric perturbation.
{That is, at second order, no new terms appear in the bias expansion of equation~(\ref{eq:generalbiaspres}) when working in full GR rather than the Newtonian limit. The GR corrections only appear in the expressions for the operators in terms of, e.g. $\zeta$. This agrees with the conclusions reached by \cite{ip/schmidt}, who argued that new relativistic bias terms should only appear starting at third order.}
We showed that the general relativistic effects do affect local clustering through the distortion of the volume element.

{{In summary, our key result is \eqref{eq:localbias}, with equation \eqref{eq:longlongGRdensity}. We described in detail for the first time how to obtain a consistent expression for the local galaxy bias model at second order  in GR. We showed how the galaxy density is related to the underlying matter density field in both the Lagrangian frame (equation \eqref{eq:localbias}) and in the Eulerian frame (equation \eqref{eq:GaldensityinX}).  Our results show in a more transparent manner that the long wavelength mode associated with the GR corrections to the Poisson equation does not modulate galaxy clustering on small scales in a similar way that the local form of the primordial non-Gaussian does~\cite{Dalal:2007cu,Verde:2009hy}, rather; GR effects deform the volume element of the local patch as galaxies evolve.  }}

\acknowledgments

We thank Marco Bruni, Robert Crittenden and  David Wands for useful discussions.
{ Some of the tensor  algebraic computations here were done with the tensor algebra software xPand \cite{Pitrou:2013hga}}.
 OU, KK  and RM are supported by the UK STFC grant ST/N000668/1.  KK is also supported by the European Research Council under the European Union's Horizon 2020 programme (grant agreement 646702 ``CosTesGrav"). RM is also supported by the South African SKA Project and the National Research Foundation (Grant No. 75415). FS acknowledges support from the Starting Grant (ERC-2015- STG 678652) ``GrInflaGal'' from the European Research Council. CC   was   supported   by   STFC   Consolidated   Grant ST/P000592/1.

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