Hey everyone, it's Missy. Thank you for watching. Today's tutorial is some easy last-minute holiday hairstyles and today's video is sponsored by Lounge so I'll be using a few of their products to show you how I would prep my hair for these types of hairstyles. Their products are also up to 70% off which is a huge savings so I'll be sure to link to those down below in the description box if you want to check them out. If you're new to my channel make sure to subscribe below and click the notification bell so you'll be notified whenever I post a new tutorial and let's get started. Before any hairstyle I always curl my hair so to protect my hair from the heat I'm going to start by spraying it with the Lounge rival heat shield. This is a great product for helping prevent damage from styling tools and since it's infused with keratin and chamomile extract it leaves my hair brushable while also adding shine which is really nice. I'm using the sienna paddle brush which is lightweight and cushioned so it glides right through my hair detangling any knots and helps distribute the product. So then I'm just going to take one inch sections and curl them away from my face holding each section for about 10 seconds. The curling one I'm using is the Lausanne 5 piece curling wand set from Lange. It comes in this convenient case and includes five different wands which makes it really easy to switch between a variety of curls depending on the look you're trying to achieve and I went ahead and linked this one down in the description box in case you want to check it out. So for my everyday curls I usually like to switch between the 25 millimeter and 32 millimeter sizes and today I'm using the 25 millimeter to create a more defined curl. This wand heats up quickly which I really love. I like to keep mine set around 300 degrees but I really love how easy it is to create an effortless everyday curl using this wand. So once all of my hair is curled I like to just use my fingers to gently comb through them so they're softer and now that my hair is all prepped I'm going to show you three easy styles that would be great for any holiday parties you have coming up. So for the first style I'm going to start by dividing off a section of hair on the right side of my head and a similar size section on the left side. I'm just going to bring these two sections together at the back of my head and tie them off with a clear elastic band. Then I'm going to take the ends and flip them up and around the elastic band so the two sides are more twisted. Then I'm just going to hold on to the elastic with one hand and then gently pull on the edges of the twists to make them appear more full. And then I'm going to hide the elastic band by taking a small section of hair wrapping it around the elastic band and then I'm just going to slide a bobby pin underneath to hold it in place. Now I'm just going to create one more section repeating the exact same steps again so I'm dividing out a section of hair underneath the first on both the right and the left sides. I'm going to tie them together at the back of my head underneath the first elastic. Then I'm going to flip the ends around the elastic band so the two sides are twisted. And then I'm going to gently pull on the edges of the twist to make them appear fuller.  For the next hairstyle I'm going to divide out a section of hair above both of my ears to give me an idea of what hair I want to use for this style. So once I see how much hair I need I'm going to let down the hair on the left side to use later on.   So then I'm going to pick up the section from the left side that I had divided out earlier and I'm going to cross this over the elastic band and then wrap the hair underneath the elastic. And then I'm just going to slide some bobby pins into the hair until it feels nice and secured and won't fall loose. Then I'm just going to go back over the style pulling on the hair at my crown just to add a little bit of texture to this hairstyle. I love how this hairstyle looks a lot more difficult than it actually is and that the hairstyle itself helps hide the elastic band.  And I'm just going to take this section and twist it upwards and then bring in a new section of hair after each turn on the bottom side. So after a twist I'm adding in a new piece from along my hairline. Then I'm going to twist the hair upwards and add in another piece and just continue working my way along the left side of my head until it reaches the nape of my neck. And then once I've added in hair from the nape of my neck I'm going to grab a clip and just clip the twist at the back of my head to use later on. So now I'm going to repeat the same steps and create a second twisted section with the hair on the right side. So another way to create the same style of twist is to pick up a section near the hairline, divide it into two pieces and I'm going to twist the bottom piece over the top while adding in a new section of hair to the bottom piece after I wrap it. So I'm just going to cross the bottom piece over the top piece, pick up a new section of hair from along my hairline, adding it into the bottom section. Then I'm going to cross that bottom section over the top creating a twist and just work my way down along the right side of my head until it reaches the nape of my neck. And then once I've brought in hair from the nape of my neck like before I'm going to combine all of the hair together and tie it off with a hair band to create a ponytail. And then to hide the hair band like usual I'm going to take a section of hair from the ponytail, wrap it around the hair band and pin it underneath with a bobby pin. And there you have it. I love how easy and elegant this last hairstyle looks. So that wraps up today's tutorial. A big thank you to Lounge for sponsoring today's video. Be sure to check out some of their items that are up to 70% off and I'll leave a link down below in the description box. If you're new to my channel make sure to subscribe and be sure to click the notification bell so you'll be notified whenever I post a new tutorial. Thank you so very much for watching and I will see you next time.


Okay, so all are able to see the screen also, the notes on the screen, okay, great. So and welcome to today's class and today we will be going through the survey and briefly summarize the book of chronicles that is the first and second. So the author of this first chronicles does not specifically name but then the tradition says that the first and second chronicles were written by Ezra. So as per the date of writing we see the completed books were probably compiled post exile period about 450 to 430 or some scholars say 425 BC probably in Jerusalem where the court records were kept. Now for the purpose of writing this book, we see the first and second chronicles cover mostly the same information as first and second Samuel and first and second Kings too. And also just like the book of Samuel as it was one single book and also the book of Kings was one single book, even the book of chronicles is actually one single book and for the purpose of reading it and making it more clear they divided the book into two, perhaps the and this book the first and second chronicles focuses mostly on the priestly aspect of the time period and also focuses on King David's rule. The book of first chronicles was written after the exile to help those returning to Israel to understand how to worship God. And we'll see also the second point there is the history focuses on the Southern Kingdom that is the tribe of Judah, Benjamin and Levi and we see these tribes tend to be more faithful to God during the rule. So the first chronicles can be turned to book of first chronicles and before we turn let's see in the notes what does it say comparison of first and second chronicles to the other books. Yeah the courses of Capetive we see here and yes this is what something that I would like to look into the declaration occurred in three stages. In 605 BC we see King Nebuchadnezzar first invaded Judah that is the Southern Kingdom and took away Joachim the King Joachim and the leading nobles which includes Daniel and in 597 BC we see Nebuchadnezzar next invades Judah and took away Joachim and leading nobles which includes Prophet Ezekiel and in 586 BC the third time Nebuchadnezzar invades Judah and this time Zedekiah the king broke his alliance with Nebuchadnezzar and had entered into alliance with Egypt to throw off the Babylonian yoke and Jerusalem fell both the city and the temple were destroyed so this time Jeremiah was taken as a captive to Egypt and we see 70 years of captivity for Judah which is a southern kingdom of Israel was predicted by the prophet Jeremiah in chapter 25 and we also see in chapter 29 and thus it happened in second chronicles chapter 36 and we also see that in the book of Ezra and there's a comparison between Samuel kings with chronicles yes but this we will move on to a book from our Bible first chronicles so the first chronicles I'll just stop the share okay so the first chronicles begin with the first nine chapters of genealogy long list of names and you will read these and think that this is kind of boring and that may be true for some of us but actually these are very important you know when we go through the genealogy we understand the line age of the descendant so the further list and genealogies are been scattered throughout the rest of first chronicles in between the book of first chronicles we also see David's ascension to the throne and this action thereafter and and we see how the shaping of genealogies emphasize on the two key line age one is the messianic king and the priesthood so the first in the line of the promise messiah king so a lot of spaces dedicated to tracing the line of Judah and that led all the way to king David to whom the messianic promise was given and then from David we see the author traces the lineup into his own day the other family line that receives lot of attention and that is the priesthood the descendants of Aaron who of course served in the temple and we see that the authors designed the book to emphasize two clear themes throughout the book first we see there's a hope that he creates of the messianic king and the second we see the hope of a new temple so here we see the author gives a strong hope again and again that the messiah there is a messiah he will come and he will come to build a new temple and it's rooted in these genealogies so that's the main reason why it's been repeated the same thing that has been stated in the book of Samuel and book of kings has been repeated in the book of chronicles now we may think why is it repeated does this also remind us of any other book that god repeated things was there any book deuteronomy yes yes yes god repeated why did god repeat deuteronomy why did god repeat the same things again god instructed moses to tell the generation what happened in Egypt i guess to remind them again yes yes lubega please go ahead i think basically because god was talking to the to another generation it was god was like telling them because we almost know that there were only two people were remaining on the former generation who had had the information before i think that was cadiban geosha so since they are going into the promised land and these were new people so that's why god had to repeat the information to them that's why it could be a deuteronomy the second of the century yes yes brother thank you that's exactly yes because they were in the wilderness for about 40 years the older generation were almost gone and the younger generation who have grown they would not even know why they are in the wilderness why they are living in the tent what is happening around them so moses tends to repeat repeat and tell the generation and remind them so god is actually asking moses to remind the generation of how god supernaturally rescued them from the 400 years of slavery so that's the reason of the repeat the same thing yeah in the book of chronicles again there's a repeat why do you think it's been repeated about her david though we have seen the stories about david and most of these are familiar from the book of samuel and kings but again there's something really different in this chapter we see that the author leaves all the negative stories about david where he portrays he does not portray about his weakness or the immoral character or about chasing david around the desert and persecuting him there or the story of david's adultery with betsheba and then murdering her husband all that is not mentioned here but then david has been portrayed as a in a very as a good king in a very positive manner and god's covenant blessing over david and this descendants so this reminds us it doesn't mean that the author want to leave away the flaws of david but then the author is mindful that it has been written in the other books which we would anyways get to know but then you know hear the main intention of the author of the first and second chronicles for us to know the faithfulness of god even though when david made mistake god is a god of mercy and forgiveness when david repented when he genuinely asked sorry god forgave david and also it also shows that god is a promise keeper when he makes a covenant he keeps his covenant yes he punishes those of sin yes he punished david's descendants and even david you know he had to face his consequence of a sin but then even the later part we see whenever the kings or the later descendants of david when they made mistake and they repented we see the heart of god the forgiveness of god the mercy of god over his descendants to keep them he does not remove them just like how he removed sol and his family but then the covenant promise that he made with david throughout we see how god keeps his covenant that through his descendants the messiah the hope of messiah will be fulfilled so we we learned something new that we find in this we don't find in the book of samuel and we only see in these book is the very positive david in a very positive light so we see in first chronicles chapter before we go to first chronicles chapter 22 i would like to share about as we study the genealogy in chapter 4 we see chapter 4 we see verse 9 and 10 a short story about a prayer of jabez okay we it does not clearly say about his is complete detail but in chapter 9 it says and jabez was more honorable than his brothers and his mother called his name jabez okay it gives a genealogy of husan ubigat home and then it gives a story about jabez that jabez was more honorable than his brothers so here's a person called jabez and he was more honorable than his brothers and his mother called him jabez saying because i bore him in pain because i bore him in pain and so in this verse 10 we see jabez called on the god of israel now what is it in the name does the name make difference as name matters class you can go ahead and tell me does name matter yes the verse is name matters yes niki niki please go ahead yes so i think it definitely matters for the simple reason even in the bible that you see paul became sold and israel and so many names are changed my understanding of why it matters is because of just the power of confession also you're confessing that over your life and you're calling that person that so that is one i think that's the main reason what you're confessing constantly and that's what happens in your life and with your permission can i just sneak in a question in between this yes yes so i've had this question for a long time about jabez that it says he bore him with pain and this is my understanding of it of why would someone go out of the way to say that she bore him with pain and the and i thought one could be because actually that maybe they were all living with so much faith that the others didn't really have so much pain but this one particular person had pain and it had she mentioned it in the word of god is it something like that or is it some other reason this was around the time when my wife was pregnant that's why i was wondering why so anyway you can answer this thank you yes thanks niki yeah unlike you we all may have different questions about you know jabez and why actually he made a prayer like this and when the mother says i bore him in pain what actually the pain mean was it the the natural pain but every delivery women goes through or was it something to do with her personal life or in the season when she carried the child jabez in her womb we actually don't know because it's not clearly mentioned we don't know this we all of a sudden this prayer just comes between but what is our learning what is the essence of these two words about jabez is what is no matter what his background was no matter i come to that point okay so i want to i i'll answer to your question i'm sure as i share it will answer because if i go to the conclusion that i will not be able to explain what i wanted to say yeah the importance of the name so why the name was imported in the jewish tradition or for some of us even now it goes back to genesis you know chapter one two where you know god gave man the power power that man is not even aware of now but but then adam knew the power because god trained adam in the garden of he didn't by telling adam he brought all the animals and he said adam named them so whatever adam named them one by one and they were called by that name and the very nature of each animal was that and they became that so we see that god was teaching adam the power was in his mouth in his mouth because god told man to have dominion over the earth and so imagine that they were a big dinosaur and a huge mammoth those days and i don't think adam was controlling them with a stick no but then adam would have just controlled them by his word he would have just spoke to them and they obeyed god you know god made adam sovereign over all the creatures of the earth isn't that amazing and same way as he was and adam was also trained by god there that to know that there was power in his words and so in the bible we see that wherever the word of king is there is power especially we read that in the book of samuel kings and chronicles and uh but but we are using how are we using our words today are we using it uh well or are we using it by speaking negative that leads to death because we see in proverbs 18 21 it says uh the death and life is in the power of her tongue so whatever we speak we have the power and later in the same in the book of genesis we see adam and eve sin and you know they they fall and uh you know we see god saying uh you came from dust and dust you shall return so what did yes that that very word was became effective in adam's life and we see but that took nine thirty years for that to take effect but at that moment his spirit died so uh we saw the process of died but this is what god said to them they have sinned and in sweat of your face you shall eat bread until you return to the ground for out of it you may take it we see that in genesis chapter 3 19 to 20 when god said that to adam he turned he turned to eve we see in the book of genesis chapter 3 if we take okay uh when god created adam and eve you know god only gave name to adam and he called both of them see when you turn to book of genesis chapter 3 i'm sorry i didn't mark and keep the verse ready but then i'll just see he named them both adam did anyone find that verse i think it is in chapter 2 out of ground chapter verse 19 chapter 2 verse 19 out of the ground of the lord god formed every beast okay and whatever adam called it not that is it genesis five two i'm just checking brother one second please yes male and female bless seven man can they created adam no no not that not that sorry i can give that verse later to your lizard okay please is it 127 man but god created where is the name adam first appeared he created man and women and and called them adam what is the verse okay i i can get that verse and give it to your sorry okay we'll get back to our lesson so god called both of them adam and we see when when god told them you shall return to dust immediately we see adam knew the sentence of death has been pronounced over both of them but he loved his vibe so he turned straight into her and said but you shall be the mother of all living and he called her hava in Hebrew in English we call it heave but in Hebrew it is called us hava you will be living and not dying and you will be imparting and proclaiming life and then so this is what adam gave name to eve knowing the power of words we need to believe you know in the power of the word that we speak into our life is it bringing life or death in the same way jabb is you know no matter what his mom named him what his mother named him okay saying jabb is that i bore him in pain okay but jabb is knowing the power on the god of israel the one true god he looks up to god he looks up to god see verse nine says jabb is was more honorable than his brothers but in midst of all that jabb is saying that i want to be a blessing blessing to your people and he looks up to god and he calls on the god of israel knowing the power that god can bless him despite his name god can change his life and make him a blessing than anyone you know when he seeks the god of israel that's why in verse 10 he prays this way oh that you would bless me in deed and enlarge my territory that your hand would be with me so you know the power on the importance of the hand of god and that you would keep me from evil that i may not cost pain so god granted him what he requested so uh the point that we wanted to make here is jabb is despite a situation despite what the people called him what his mom named him okay what people looked at him you know he called on god of israel and when he called on the god of israel god answered his prayer request god responded to him and his prayer was answered and we also see the foreshadowing in uh first chronicles chapter 16 verse 33 that david's song of thanks giving to god and he refers to the time when god will come to judge the earth and this foreshadow we also see in matthew matthew chapter 25 in which jesus describes the time when he will come to judge the earth and later part we see through the parables of the ten virgin and the talents he wants that those who are found without the blood of christ covering their sin will be cast into utter darkness he also encourages his people to be ready because he's coming and we also see uh jesus demonstrating like he will separate the sheep from the goats and time of judgment so part of this davidic covenant which god reiterates in first chronicles chapter 17 from 13 to 14 can we turn to 17 yeah 13 to 14 we uh or from 11 onwards we can read and it shall be when your days are fulfilled when you must go to be with your fathers that i will set up your seed after you and who will be of your sons and i will establish his kingdom he shall build me a house and i will establish his throne forever i will be his father and he shall be my son and i will not take mercy away from him as i took it from him who was before you your god is talking about sol and verse 14 he says i will establish him in my house and in my kingdom forever and this throne shall be established forever so here we see it refers to the future messiah who would descend from the line age of david describes the son who will establish in god's house and whose throne will be established forever this can only refer to jesus christ so we see uh in the genealogies as the ones and first chronicles may seem very dry sometimes we may think it's not important but then time and now we see god reminding his children about the the descendant the line age so that we see in matthew chapter 10 30 we see that god knows each of his children by their personality even to the number of hair on our head so from this fact that we can take comfort that you know uh we are in god's mind god is mindful of each of us and also in uh we also see the confirmation in the book of revelation chapter 13 verse 8 it says we belong to christ and our names are written in the book of land of life lambs book of life so we see that god is faithful to his people he keeps his promise and he brings the descendant that's the main reason time and again we see god uh you know the authors are mentioning the line age very very important because god promised that there would be a messa messanic king coming from the line age of david so we need to know through whom it is coming and how it's coming and throughout the story we see god is a covenant keeper he's a god of mercy god of forgiving he forgives us when we repent and you know he blesses us and also uh we go back to verse chapter 11 verse 1 2 3 we see the fulfillment of god's promise to david when he made king all uh you know overall is ryan we we can be sure that his promise to us will be fulfilled no matter what promise god has made though at times that taris button the book of abacus 2 we see that write down your vision write down your promise and there will be a due time without any tari it will be fulfilled don't worry even when it is getting delayed but a god who promised he will fulfill but he will bring it to come to pass so as he promised a blessing to those who follow him who come in christ in repentance and who obey his word god will bless them so that is the learning we see in the book of samuel in the book of kings and in the book of chronicles we see that obedience brings blessing and disobedience brings judgment so we may have a question okay we went through all this in the book of samuel and the book of kings and again repeat of david's story but here we see the positive side of david has been repeated as david is the ideal king in order uh in order to you know the type of messiah the future messiah would be from the line of david it would be very similar to how jeremiah or ezekiel speaks about the cupic messiah as a new king david we will study when we cover those books later on so uh this is most clear how the author retells the story of god's covenant promise to david in the book of chronicles especially in the first chronicle when we compare the story which is parallely to second samuel chapter seven we see the author of chronicles highlighting that david uh or solemn or any other king when uh when not the king the messiah which they are talking about but it is the messiah is going to come from this lineage and we know who's that messiah king now but then back then they had a hope that there is a messiah king who will come from the lineage of david so as we study these stories about david and solemn and his son and his son and their sons and their descendants lineage it gives us the hope for the future for the people at that time who were studying over reading and they were waiting upon the messiah whom the prophets have been prophesying or continuously talking about it gives them the hope that the messiah will come from this lineage so with this we will move on to second chronicles uh yeah we see uh before the second we could start the second chronicles and the first chronicles we see how david's praised god and you know passing off david to solemn and solemn has been anointed as king and the second chronicles starts with the wisdom of solemn and we see there's a lot of new things we can learn about these kings from david's lineage how the kings were obedient to god and how their obedience led to success in god's blessing and at the same time we also see some some stories about the kings who were unfaithful to god they didn't follow the Torah or they led Israel to worship idols and these kings you know they were they had to face horrible consequences because you know they defiled the temple and they led the people of Israel to worship idols and child sacrifice was happening in those days and all these things you know stirred the anger of god and you know they were led uh this led to israel's exile and we see the we saw that how king nebuchadnezzar invaded israel three times and then later egypt so when the whole section of repeat becomes a series of chapter study when the author wants later generation of israel to learn from their family history and so they become faithful to their god and to keep up the Torah we see some of the kings here in second samur second chronicles the story of asa we see how he obeyed god and how god blessed him god gave him peace all around the place just like solemn and sorry we also see uh king joshaphat story in chapter 20 can we turn to chapter 20 i would recommend as always okay it's just a survey so we cannot cover each and every chapter or every king's story but then i would request you all to please go through this book and go through each and every story and for now i we will just read one of them that is king joshaphat who was from the line age of david what happened so we see in chapter 19 then king the joshaphat the king of judah returned safely to his house in jerusalem and you know we go through it and uh he obeyed god and he followed the god of israel and for every water every battle that came up he used to inquire with god through his prophets and then step into it and we see in chapter 20 there were three kingdom amen moab and mounts here came against came against joshaphat so what happened now verse three chapter 20 verse 3 we see that joshaphat feared and set himself to seek the lord and proclaimed a fast throughout all judah and judah that is the southern kingdom of israel and judah gathered together to ask help from the lord and from all the cities of judah they came to seek the lord and we see joshaphat stood in the assembly okay uh by assembly and from all the cities of assembly of judah and jerusalem in the house of the lord before the new court and said oh lord god of our fathers and you are you not god in heaven and do you not rule over all the kingdom of the nation and in your hand is there not power and might so that no one is able to withstand you he just prays and he gets a he gets a message through his prophet that god they come and say that you know the battle belongs 14 then the spirit of the lord came upon jehazel the son of zakariah the priest the son of benaiah okay he says listen all you of judah and you inhabitants of jerusalem and you king joshaphat thus says the lord to you do not be afraid not dismayed because of this great multitude for the battle is not yours but gods so tomorrow you go down against against these three kingdom and god will fight for you and verse 17 you will you will not need to fight in this battle but position yourself stand still and see the salvation of the lord who is with you oh judah and jerusalem do not fear nor dismay tomorrow go out against them for the lord is with you so joshaphat believes on this promise whatever they said and what he does is we see he puts the singers the men who praise god before in the battle usually how do they prepare for the battle they they would prepare with a mighty man of valor they put the strongest man uh in front but then joshaphat prepared this battle in a much different way because god said through his prophet that the battle belongs to god okay battle belongs to god so you know he stands strong with that word and he puts the the musician the one who praises god and the friend and you know the others later and he says and they sing like this they go marching to the battlefield saying praise the lord for his mercy endures forever verse 21 which is a very powerful praise let's say which the king raises the praise to the lord and verse 22 now when they began to sing and to praise the lord praise the lord said ambush against the people of amen moab and mounts here who had come against judah and they were defeated so what happened we see the power of praise the power of praise so just like joshaphat if we have such uh you know a huge uh situation before us what would we do would we seek man's help how would we seek god whose help do we seek in our life if any circumstance as such arises can be financial it can be health it can be our carrier it can be a personal life it can be any of the situation that could arise as a big mountain at a big battle just before us like how it has suddenly for joshaphat where he was not prepared how would we face anyone in the class can unmute and answer so just like joshaphat we need to we need to inquire with the lord we need to first bring it to the lord even if before we could take help from any man we need to bring it to the lord there's nothing wrong in taking godly counsel or getting the medical advice seeking doctors taking medicine or taking help from others despite our situation or what our situation may be but then first is we need to bring the case to the lord we need to inquire with the lord so that the lord can deliver us and when we praise in our situation praises pleases god so that we don't have to fight the battle but then god takes it god fights our battle and the victory belongs to us just like how god fought the battle for king joshaphat he don't have to by the time they reach the battlefield he saw everyone dead in the later verse we see every one way dead not even one man was alive and it took three days for them to take the spoil because the goods that these people carried for the battle was huge and it took three days for them to gather the spoil and come back and god gave them victory and the you know and the king joshaphat and his people rejoiced over the victory what god gave them and they praised god they made sacrifice and to god so just like that when we bring our situation out our battle front of god it can be anything god gives us the victory god gives us the victory when we seek him he is the same god who's who was then and now and he's the same god who answers our prayer he is a living god now the book conclusion is very unique at the very end of this book we see the king of persian and his name is syris and he tells the israelites that they can go back home return from exile and he's also not only sending them back home but he's telling them we build the city and the temple and here he says you know the last time very abruptly ends the last verse chapter 36 verse 23 it says whoever there is among you and all of his people may the lord is god be with him and let him go up and let him go up and that's how the book ends with the incomplete sentence now of course the author knows that or the first return from exile and the story of Ezra and Nehemiah but clearly in his view the prophetic hope of israel were not fulfilled in those events and so this incomplete ending shows that the author hopes is set on yet another return from exile when the messiah will finally come to rebuild the temple and to restore god's people so the book of chronicles we see that there is a hope hope that god still keeps his covenant promise god still shows mercy and god yet rebuilds his temple and his people and we also see that god will yet come in great sovereign saving mercy and great towards his people so the book of chronicles as well as the book of Samuel and kings we see the pattern of sin repentance forgiveness restoration of the nation of israel and in the same way we also see in the in the New Testament in 1 John 1 9 that god is patient with us and he forgives us our sins when we come to him in true repentance we see that throughout the book how god forgave the kings whoever repented the same way we can take confidence and comfort in that fact that you know god years our prayer he understands our situation and he forgives our sin when we repent genuinely he restores us to that fellowship with him to that intimacy with him and sets us on path of joy so this is what our learning would be from the book of chronicles that covers both the first and second chronicles when we study each king's life the one who disobeyed god and act to face the consequence of the very own act and for the kings who obeyed god we see the hand of blessing on them and there are certain kings yes they disobeyed initially but they learned from the mystic when they repented we see the mercy of god for giving them and restoring them back and then we see the blessing of god and their life so this is what the first and second chronicles is all about so now I open to the class you would like to share anything you can add on from what we studied or from the book of first and second chronicles if I've not covered yes I've not covered many aspects but the main points I touched upon is there anything that you would like to add on it would be a blessing to us as a class anyone in the class what was our learning anyone yes brother please go ahead brother please go ahead I wanted to add something about the story of Jabez when when you read in the extra books of the bible the Mishina when you try to read in the Babylonian Talmud and when you try to read also in the Jerusalem Talmud you realize that this madame called that boy that name not basically because of the natural pain she had in giving birth to because it automatically every mother has the same issue but basically she called him this name because it was an outcast for a child to be called by to be referred to by his mother's side so since this boy wasn't referred to by the names of his dad it means that this boy was given birth when the mom was at their home which was a mischief which was not right in their family or in the Jewish culture hence saying that I bore him our pain thank you so much thank you brother thank you so much Nikki initially you asked for a question did what's your question answered yes yes thank you brother Lubey thank you yeah any other learning I open to the class you would like to add on to the point anyone? Zellie, Divya, brother Subashish, anyone in the class? Georgia, Hamilton, but Isaac, Jeffina since a long time you've been very quiet please go ahead I just called out a few names it can be anyone from the class Alan or said okay as the time is up okay but was the class helpful will you be able to understand it we learn something new the class aside and am I audible I see a comment okay it would be nice if you can verbally say so that that will be helpful I see a lot of comments coming up okay praise god we learned a lot of new things and it was a blessing for us in our spiritual work of life praise god thanks Sid thank you ma'am for sharing this knowledge praise god okay okay thank you okay let's end this class with a word of prayer dear god we thank you we praise you we love you we honor you you are our god you are our lord the god who where then is the same god who's with us now you are the true living god the god of Israel you're the god of Abraham Isaac Jacob David the same god is alive and living in midst of us you're the god of understanding you're the god of forgiveness you're the god who keeps the covenant who makes the covenant and you keep the covenant you're the god who's faithful lord today I lift up each and every student we're attending this class now and later lord I pray and I thank you that you have called each one of us and you are faithful to that calling lord you will strengthen us in the way in the path that we walk in our father during the difficult times lord as we seek you I pray that you will be with us you will strengthen us you will answer our prayer just just the way you answer Jabez pray you will be with us just the way you were with king David and you led him you blessed him and his descendants same you will bless us lord we also pray for the greater wisdom of father the way you're blessing Solomon same way you will bless each of us with your wisdom the wisdom of Jesus as we read in the book of New Testament saying that Jesus grew in wisdom knowledge and understanding lord I speak that over each one of us in our class and we'll listen later we'll watch this class later that we speak that your wisdom will be increased in us lord you are the source of all wisdom thank you father thank you lord that you are comforting us and you're leading us that you are the god of hope that no matter what we face in our life the battle belongs to God when we when we bring it to you when we inquire with you you are the god will bless us you will fight our battles and you will give us the rest and you will give us the victory and our hope is in you we claim this over each one of us lord in Jesus mighty name we pray amen amen class thank you so much for joining in today's session god bless thank you thank you thank you good bless thank you thank you god bless thank you thank you good bless


Quantum field theory in a fixed background. I tried to emphasize the two main new things about a curved background, namely the ambiguity of a vacuum choice and the phenomenon of spontaneous particle production from the background. And this is the mechanism that inflation, in inflationary backgrounds, produces the fluctuations that we see in the CMB and hope to see even more of. And the last two lectures will be about things we haven't measured but hope to measure, which hopefully will give us much more information about the nature of inflation. So a brief recap. So the inflationary Lagrangian that we've been studying is probably the most vanilla model of inflation, just single field inflation with potential that I just required that it generates the inflationary solution, which is a background that roughly speaking looks like the sitter space and for which the slow row parameters are small enough that I mean the slow row approximation and can use the sitter mode functions to describe the particle production. And I explain this schematically, but just because the calculation itself is hard, it takes time. I'm not going to be able to do it in real time. We would take the four lectures to derive the power spectrum and a bit of what we're going to do today. But the idea is because GR is a system with constraints, we specify a spatial metric on a three slice. And then these two other quantities are Lagrangian multipliers. They don't have dynamics. So we should solve for them perturbatively and plug them back into the action. So that's how we got the quadratic actions. Or I just told you the result. We didn't actually do the computation, but I recommend that you do it. It's educational. And the result looks simple, but to get here takes a bit of work. And this is the power spectrum. So it's essentially the power spectrum, the two-point function of a massless scalar field in the sitter space. The only key point is that the normalization factor appears here. And because the normalization factors are different for scalar versus gravitational fluctuations, there's a separation of powers. The scale that separates the amount of power in gravitational waves versus in scalar fluctuations. And so that's how we explain the fact that we have seen we've measured only these guys here and not these guys here. So today is all about the next correction to these results. And I'm going to forget about tensors today, at least gravitational waves. So last lecture is going to be all about gravitational waves. So today, I want to do everything I can about scalar fluctuations. So this is the plan. I had to shrink quite a bit the things I wanted to discuss. I guess I was too much stuff, so I hope I can go through these things with you guys. So the first idea is to describe self-interactions of the inflaton, of this zeta fluctuation. So I'll describe. So this was first done very carefully in a paper by Maldacena from 2003, 2002, 2003. And it's a very nice calculation, but the result is undetectable. It's like tiny results. And I'll explain to you, I'll try to explain schematically how we are able to build a theory that enhances the size of the three-point function. Then I'll discuss a particular limit of the calculation in which it's a consistency check. We know what the answer has to be. It's a bit like the Weinberg soft photon theorem, but here for inflationary fluctuations. In fact, it's not a bit like. I think it's essentially the same thing. And then I'll use that as an excuse to say a few things about the SCFT. And finally, I'll talk about some recent developments, the idea of using these inflationary correlation functions as a particle detector. So this slogan is cosmological collider physics. So the idea is that inflation happens at very large energy scales. So you could have excited particles that we don't have access to in our colliders. And they would leave imprints that cannot be mimicked just by self-interactions of the inflationary fluctuations. So the idea is how do we detect these things? So that's the plan. So the first thing is we want to discuss. So we calculated the free theory, just the two-point function of the scalar fluctuation. So today's all about the three-point function. So this is the observable that we're going to compute today. And it has many names, the literature. So it's called bi-spectrum, which is a little bit misleading because there is a bi, but it's three fluctuations. And the four fluctuations is a tri-spectrum and so on. I don't know why. I guess the power spectrum is considered the unit spectrum and this is the bi-spectrum. This is, of course, a three-point function. And the final name is non-Gaussianity. The reason it's called non-Gaussianity is because if you have a Gaussian stochastic process or free field theory, because of the weak theorem, everything is expressible in terms of the two-point function, so in particular, the three-point function should vanish. So a non-vanishing three-point function is a sign of non-Gaussianity of the stochastic process. One thing is that remember here that momenta have to add up to 0. And that's just because of the isometry of the background. So the background is, I'm going to use approximately the sitter space. So it's something like this. Background solution and then I have a translation isometry. So I can talk about Fourier modes and they have to add up to 0. So I have three Fourier modes here. And I'm going to put the prime just not to write the delta function continuously of momentum conservation. So they have to form a triangle. There's even an article in this quanta magazine called Triangles in the Sky about this type of non-Gaussianity. So it's kind of a nice article. Have a chance to read it, take a look. So let's see. What should I tell you first? First thing I want to tell you is how to, there's a bit of a filter definition that I'm going to do, but it's to make the computation of the cubic action easier. So let me draw a diagram to explain how we're going to calculate this thing. So the power spectrum, zeta, zeta, a way to see how we're calculating is there is the beginning of inflation here, eta goes to minus infinity. And then we're imposing the condition that we're in the vacuum. And then there's spontaneous pair production. So the pair is produced. And then they just start moving in opposite directions because of expansion of spacetime. And then here at late times, eta going to 0. Their fluctuations freeze out. So this is the diagram that gives me the power spectrum. So time is going in this direction. For the three-point function, zeta, zeta, we're going to compute this diagram. So I have some points here. There's spontaneous creation of three particles. And then they separate. This is the diagram. So to calculate this, of course, I need the cubic vertex. So that's our first task. So to calculate the cubic vertex, we need a little bit of a tweak here. It's just to make the calculation simpler, to make the inverse of the metric simpler to work with, and so on. It's a redefinition that was actually proposed in the 90s by Bond and Salopex, I think. So we're just going to do a bit of a redefinition. And this is a technical comment. But it's everywhere in the literature. So instead of 1 plus 2 zeta, we just see that as the first two terms in the expansion of an exponential, just so the inverse matrix, the inverse three metric is easy to write down. So now once we have this, we're going to stick it in the action. So let me describe the procedure in words, and I'll just tell you the answer. So the procedure has quite a few steps. So first thing you do, so the goal is to find the cubic Lagrangian for the zeta fluctuation. So I'm going to stick in the three metric in the action. And I'm using the fact that phi is phi bar of t. Remember that there is no scalar fluctuation. Then n and ni are going to be Lagrange multipliers. I'm going to solve these constraints equations. And there is a neat argument by Maldesena that even though I need the cubic action, I only need to solve these constraints to leading order in the fluctuations, which I had to do for the quadratic action. So I don't have to solve again the constraints equations. So then I plug these back in the action. And then I work hard and get the cubic action. And the cubic action is a mess. And then there is so let me solve constraints, plug it back in the action, expand to cubic order. And there is a final problem with the cubic action that is the following. So you will find terms in the cubic action. So remember that here there is an epsilon in front of the quadratic action. It's because I can't access this gauge if there is no inflationary background. So there is no Higgs mechanism when I'm at the top of the potential when symmetry is restored. So the fact that I picked the clock allows me to eat the fluctuations of the scalar field into the metric. So the epsilon appears here. But somehow it doesn't appear in the cubic action. So there are terms that are naively not slow roll suppressed. But then when you do the calculation, they don't contribute to the three point function if I assume that the mode functions are those of the sitter space. So what one needs to do is to ensure that one is using the cubic action, just the leading order in slow roll term. The reason why, I don't know if I'm making myself clear. So let me write it down. So S3 is going to have a term that is order epsilon to the 0 plus some term for the epsilon to the first power and then order epsilon squared. What happens is, so I'm in the slow roll approximation. So I look at this and I'm like, oh, life is easy. These are higher order in slow roll. I just throw this away. And then I use the sitter mode functions and do the calculation of the three point function here. I'll get 0. What does it mean? It means that I use the wrong approximation. I use the sitter mode functions. But I'm in the inflationary background. I have to find the quadratic mode functions next to leading order, which is the first slow roll correction to the mode function. So then I'll get some epsilon to the 1 term. So I look at this and I say, OK, so now my mode function, zeta classical, so I'm using interaction picture. And I need the mode functions at quadratic order. And we are using the leading order in epsilon. Let me put the root epsilon here in front of the classical mode functions. Let me write them down here so that you these are the mode functions. But this is in the sitter. And I'm just adding the factor of m-plank root epsilon here. So I'm assuming that it's the correct mode function. But in fact, there will be corrections that are slow roll suppressed. So there will be some order epsilon to the 0 plus order epsilon plus order epsilon squared. So if I use these mode functions and stick them in here, I get 0. So it means I have to go to next to leading order. Now I should use this correction, stick it in here, and the sitter mode functions, stick them in here. I get 0 again. So the first non-leading correction is here. And then Maldacena figured out a beautiful trick of using the sitter approximation. And at the same time, taking into account the contributions from these two terms without knowing these mode functions beyond the sitter limits. So you have to do further definitions and play a sort of complicated game with the action. And I just can't stress enough, you have to read the paper and do the calculations just because it's a tough calculation. At the end of the day, there is a fielder definition. It's not the original zeta. I take zeta to some f of zeta. And this fielder definition essentially erases these terms here. So S3 written in terms of this f of zeta is going to be explicitly order epsilon squared. Epsilon squared times something else. And this is the right size of the cubic vertex. It has to be slow row suppressed at least. So you would imagine that at least it's epsilon to the one. But because everything is a weak coupling here, you would imagine that the coupling constant should appear more than it appears in the quadratic action. It turns out it does. But if you write the naive cubic action, it doesn't. So it requires quite a bit of work to get to this point. Sorry that I'm not being explicit. This is a long computation. So once one lands in here, then I can just proceed to do this calculation. So after all of this work, I tell you that there is a single term in the cubic action that we need to worry about. And the term is of this form. And it will look a little bit weird. But trust me, it's correct. a to the fifth epsilon squared h. I set m-plunk to 1. Sorry, I forgot to see what's the theta dot. Then there's an inverse Laplacian, theta dot squared. And let me call this thing here zeta c. Because it's not the original zeta. I had to do a few field definition. So this is the action. You might be worried about this one over Laplacian here. So it's no local. But if you do electrodynamics in the Coulomb gauge, you'll see that these non-local terms are induced if I do perturbation theory. It's just because we're trying to remove all the gauge redundancy of the theory. And the price we pay for that is that locality is not manifest in the action. Technically, the way that inverse Laplacian appears is through this guy here, ni. So when I solve for ni, I have a scalar variable zeta. And ni will be roughly speaking of this form, di zeta. Something like this. I have to solve an equation where the Laplacian of ni is given by di zeta. So then ni will be given by inverse Laplacian times di zeta. Yes, spatial Laplacian. So here I'm explicitly breaking time-space symmetry. So all of these partial derivatives here are spatial. So this is a problem. So here, if you're an experimentalist, you would just give up. Because it's a problem. It's a tiny signal. And I'll give you some numbers in a little bit. And there is other stuff here that is even more subleading. So now I give you some cubic vertex. And I have to calculate this three-point function here. The way to do it is through something called the Keldisch technique, or Schringer-Keldisch method. So when I'm given interaction vertex, we learn in quantum field theory to use Feynman perturbation theory. But in Feynman perturbation theory, what we're really computing is some in-out correlation function with some time ordering, maybe, divided by in-outs. So this is Feynman perturbation theory. And if the vacuum is stable, there is no particle production, out is just a phase times in. So the phase upstairs cancels with the phase downstairs. And Feynman is essentially indistinguishable from Schringer-Keldisch type of perturbation theory. But here, the vacuum produces particles. So we need to be careful. We actually specify the initial states. And then we calculate an expectation value at late times. So what we want is in zeta, zeta, zeta, in. There is only a single time evolution operator when I do this calculation in Feynman perturbation theory. So the path integral is doing time evolution from minus infinity to plus infinity. Here, I have to do time evolution twice. I evolve from minus infinity, the initial states, to the time at which I'm inserting these operators. And then I have to produce the ket. So I have to evolve backwards in time. I evolve from time t back to minus infinity. So this requires a slight modification of standard perturbation theory. It's called Schringer-Keldisch, or in-in perturbation theory. So the path integral actually doubles. You need to talk about twice the number of fields. And it was used by a condensed matter physicist quite a bit. And I think it's described very neatly in a paper by Weinberg. So there's a paper by Weinberg in which he actually tried to compute the one-loop correction to the power spectrum. And then he wanted to develop this in Weinberg-esque detail. So you should take a look at Weinberg 2005. He has a whole appendix in which he describes this in a lot of detail. For us, because we're just interested at three level, it's very simple. It's a mild modification compared to Feynman perturbation theory. Let me just explain to you the results. Get my notes so I get the right factor. I'm going to be a little bit abstract, but I have a state psi. And I have an operator O that I'm computing at. I want to know its expectation value at time t0. And then I want this correlation function here. So we're doing interaction picture. So we start with a free vacuum. And then we're going to do time evolution on both sides. So we need the Dyson time order exponential interaction Hamiltonian. So this is standard. But now I do time evolution from minus infinity until time t0. Then I evolved the vacuum states all the way to time t0. Then I seek in the operator O. And now I have to evolve. I have this produced the ket. Now to produce the bra, I need to take the dagger of this thing here. And then I'll get anti-time ordered e to the plus i integral minus infinity to t0 of the interaction Hamiltonian. So it's a little bit different from standard Feynman perturbation theory because now there are these two time orderings here. So in Feynman, what you do is you have your operator insertion here, let's say t0. And then you do time evolution from minus infinity until time t0 and then from t0 to plus infinity. This is the time evolution operator from t0 to plus infinity minus infinity t0. In Schringer-Keldisch, we go all the way to t0, and then we come back. Because we're only interested in tree level, we just need the first term in these exponentials.  So that's the difference between using Feynman or retarded Green's functions at tree level. So for tree level, we get this psi t0 psi cos minus i 0 minus infinity to t0 h inverse t. Final thing is the commutator is just coming from this guy here, added to this guy here, that's what gives me the commutator. Final thing is that there is an i epsilon prescription. So the integral, when I go to minus infinity, recall that the mode function has this exponential here. So when I go to minus infinity, this thing oscillates very fast. And then I'll use the Feynman i epsilon prescription and the time coordinate just to damp this exponential here. OK? So that's it. And then, so I know the cubic action. I know the prescription. And I know that once I do this calculation, I have to undo this field redefinition because zeta was the actual variable that is related to the curvature fluctuations. So I have to do all of this work and you get some results for this three-point function here. So the result is usually called the bispectrum. Oops, b of k1, k2, k3. And OK, it's a rational function of the momenta and it has low roll parameters in front of it and it's tiny. If I write it, it's not going to give you any insights. But you can look at Moldeson's paper for the specific shape of this thing. What happens with this object is that one thing that gives you some insights is the following. So recall that I have a triangle. So now in the case of the power spectrum, the only thing I could dial was I had the segments. So because of momentum conservation, I just have a length scale to work with. So the only thing I could do was dial the size of the wave number. Here, and scale invariance tells me that it doesn't really matter what the size of this wave number is. The red tilt just tells me that there's a mild damping of the power spectrum as I change the wavelength. So now here I have much more freedom because I have a triangle so I can dial not just the size, expand it, but I can also change the shape. So when you read the literature about this thing, people talk about different shapes of the triangle. And I'm going to contrast two different shapes, the equilateral shape versus the squeezed shape. So this is called equilateral versus squeezed. Now here is some physical information about the answer. In single field inflation, squeezed is 0. So in the squeeze limit, you get something that is essentially 0. And so this bi-spectrum peaks around the equilateral. So bi-spectrum peaks, it puts most power in the equilateral shape. So that's interesting because if I measure this bi-spectrum and I go to the squeeze limits and I see something, then it's not single field inflation. I ruled out single field inflation. All and OK, I did one example of single field inflation. But this is actually a universal. It's kind of a symmetry statement.  So this is true for all single field inflation models. This was pointed out actually by Paolo Creminelli here at CTP with Matias Adariaga in 2004. So this squeezed shape is actually smoking gun of something other than single field inflation, which is nice. And actually it's the shape that is mostly is better constrained by experiments. I don't know why. I'm not a specialist on this. But the equilateral shape is much harder to constrain. But as I said, because of the extra factor of slow row, this is essentially hopeless for single field inflation. Let me explain to you how we quantify, whether it's hopeless or not, and give you a sense of numbers, things that we can actually measure and things that are beyond our dreams. I'm going to write the way we quantify sizes of three point functions. And then I think it would be a good time for questions, because I'm going to switch gears a little bit. So quantifying non-Gaussianity. So if you open Planck's paper on non-Gaussianity, you'll see this thing here, FNL. This is how people associate a number to non-Gaussianity. So non-Gaussianity is a triangle shaped function. But then for a given template, so this is a complicated function, for a given template, people associate a number to it. Because when you actually try to look for this in the data, I don't think that we have enough statistics to be able to actually see the precise shape.  The way that people put constraints on non-Gaussianity is by this number FNL, which I'll show you the definitions a little bit weird. So I'll take my bispectrum, my three point function, K1, K2, K3, and it doesn't matter whether it peaks in equilateral or squeezed, I'm gonna calculate it in the equilateral shape. Of course, this function can be evaluated for any shape of the triangle. So I'm gonna evaluate it for the case of an equilateral triangle, and then I'm gonna divide it by the power spectrum. So this is, there's a single K that enters the game, and then I'm gonna calculate the power spectrum at the same value of K and square it. So this is a kind of historical definition of a quantifier of non-Gaussianity. It was, I think, first proposed, I think by Komatsu and Spargel. But if you just count zetas here, it seems like a bad idea because each zeta is around the 10 to the minus five, right? So FNL being large still means that this three point function can be relatively small. So you see bounds on non-Gaussianity, relatively large numbers, and you'll be, wow, the experiment is terrible, but actually it's just because there's an extra 10 to the five here. So for slow-roll inflation, which was Maldesena's calculation, FNL, this shape function peaks around the equilateral configuration, but FNL is of order low-roll factors, epsilon. Eta. So even though I'm paying an extra 10 to the five here, the size of the non-Gaussianity is probably, or the 0.01. So this, okay. I'll explain things that we, I'll mention things that we use to measure non-Gaussianity and there's some dream that there's a way to measure this using 21 centimeter radiation from the dark ages. That was on the news recently, there was some claimed detection of the of the monopole of this radiation. So this is tiny, but there are ways of enhancing this within single-fueled inflation and probably the most popular way is by, so let me say it's possible to enhance non-Gaussianity in single-fueled inflation. And the way to do that is by making the scalar fluctuations badly break Lorentz or the Cedar symmetries. So the way to do it is by giving theta a small speed of sound. So the only way I know how to explain this is by writing the Lagrangian. You'll see that if I, because of the non-linearly realized symmetries, the interaction vertices, they are, so the vertex that controls the size of the speed of sound, there will be a, sorry, so the term in the Lagrangian, I write some Lagrangian for the zeta fluctuations, okay? So there will be a term that I can change that will control the speed of sound. What happens is that when I dial this term, it doesn't change only the speed of sound. It doesn't just dial the quadratic zeta action. It also dials the cubic zeta action. So when I decrease the speed of sound, I enhance the size of the cubic vertex. And this is because of non-linearly realized symmetries.  And there's a nice way of seeing how that works, but I'm not gonna do it, unfortunately, I don't have time. But I can give you a reference where this is done. So let me just give you the size of FNL when the speed of sound of the scalar fluctuation is small. Okay, so by speed of sound, let me just write down the formula. By speed of sound, I just mean that the quadratic action for zeta is gonna have some detuning. So it's gonna be something like one over Cs squared, zeta dot squared minus cubed here minus a zeta. So when I give some speed of sound to the zeta fluctuation, it just so happens that there will be a cubic term here, zeta dot cubed or something like this that goes like one minus one over Cs squared. You can't just dial this guy here and get rid of the cubic term. That's the beautiful thing about the fact that symmetries are being non-linearly realized. And so when I make Cs small, I enhance the size of the cubic vertex and non-gaussianity is large. So that's probably the most popular way of enhancing non-gaussianity in single field inflation. So then FNL in this case will be order one over Cs squared. And actually the current bound on Cs is not very strong. So Cs has to be greater than or equal than 0.024. So this is the current Planck bound. So there is a lot of room for a crazy small speed of sounds. The reason why the bound is weak is because the non-gaussianity peaks in this equilateral shape and just a constraining equilateral non-gaussianity is hard. So that's the story. But before I continue, are there questions? No, because the Cs will appear in the power spectrum. It's just some extra parameters. So now you have like some absolute Cs. So now we're not measuring H over M Planck times epsilon, we're measuring this thing here. So because I introduced an extra factor, I need to do more experiments to disentangle things. What one can do is that there's an extra term that will kill this thing. But then I have to start doing a lot of fine tuning to make this non-gaussianity disappear. So if you don't have a fine-tuned theory and the speed of sound is small, you will generically have large non-gaussianity. And the bound is pretty weak. It's just because equilateral non-gaussianity is hard to constrain. Well, a squeezed non-gaussianity is in much better shape. And do I have here... I don't have the current bound on local non-gaussianity, but I can bring it tomorrow. Let me explain... Yeah, you can, and this thing here is coming from some effective field theory that is essentially parameterizing to leading order and derivatives, all sorts of cubic terms. So I'm not doing a single field inflation with some extra terms. So actually to make the speed of sound small, one needs to forget about the background theory. So one needs to build an effective field theory just for the fluctuations. Forget about the... So okay, let me say it in a different way. So the reason why in slow-row inflation, the non-gaussianities are small is because we are controlling both the background and the fluctuations with weak coupling expansion. So we're paying extra prices of slow-row. If one builds an effective field theory just for the fluctuations and forgets about where the background evolution comes from, then I can enhance non-gaussianity. So that's part of the reason. I can get rid of some of the... I can get rid of the factor of epsilon. So now I can make non-gaussianities be order one and there's the extra one over CS squared when I make the speed of sound small. So this type of effective field theory just assumes that it's single field inflation. So it will include something like star-beam-ski inflation like R squared. So let me describe a bit the squeeze limits because then I can make a contact to soft limits and tell you a bit about the ACFT. I guess I'll leave cosmological collider physics for tomorrow. So the interesting thing about B, this bi-spectrum, so there are two interesting facts. So I'm gonna write an expansion around the squeeze limits. So I'm taking K1, K2. K3, K1 much smaller than K2 and K3. And then I have a, I'm gonna write it like this. I'm gonna write it as the power spectrum of K3. So the limits as K3 goes to zero of the bi-spectrum is gonna be given by P of K3, power spectrum of K1. One and then some n from zero to infinity of K3 over K1 to the n times an. So this formula is already a bit non-trivial because I'm assuming that there is a Taylor expansion of this squeeze limits. But just believe me, any single field model will have this Taylor expansion and all the exponents are integers. The interesting thing is that all single field inflation models, A0 and A1. So you only start seeing the difference between what single field model you're working with at A2. A2 carries new information. So notice that this bi-spectrum, I'm writing it just in terms of the power spectrum. So in principle, if this is true for any single field model, and actually maybe let me write down what the answer is. So A0 is one minus NS, the scalar spectral index, so it's tiny. Okay, remember that this is a 0.04. And A1 is, okay, I don't quite remember but it's something like a D1 minus NS. It's also essentially negligible, okay? So these two terms are essentially negligible and moreover, they're all determined by the power spectrum. So in fact, by knowing the power spectrum, I know the first two coefficients in this squeeze limit. So it means that it's giving me no new information, right? I know the power spectrum, I know the bi-spectrum to leading and next to leading order in this expansion. So this is exactly like a soft theorem. I can determine the endpoint correlation function by knowing the N minus one point correlation function. And there are versions of this for as many legs as you would like, okay? So for four legs, for example, I can take one of these legs here to be very small or I can take the diagonal to be very small. So there are all sorts of soft theorems related to this. And I think they're identical to the Weinberg, the details differ a bit, but they are identical in spirits to the Weinberg soft photon theorem. A2 is really where new physics kicks in. And that's also where cosmological collider physics will appear in this expression. So, but I'll do it tomorrow. I want to explain a nice way of seeing where this thing comes from. And so the standard proof of this soft limit is the following. So in this limit here, what's going on? So this mode, the one with a very small wave number, it has huge wavelength. So it goes outside of the horizon way before the modes of wave number K1 and K2. Okay, so recall that we are computing diagram like this. But now we are in a situation where one of the modes exits the horizon way, way before the two other modes exit the horizon. And now if you look back at what I raised, but just Gij was something like a squared e to the two zeta, delta ij. When this guy freezes and is way outside of the horizon, I'm essentially again in the original background. If I put some zeta long here that has no time dependence and whose wavelength is enormous, this is just some numbers just rescaling the coordinates. So it's as if I was doing the calculation of the power spectrum in a different coordinate system. So that's the intuition. So that's why in this limit, I'm actually kind of remeasuring the power spectrum or the bi-spectrum is completely determined by the bi-spectrum. So this is the standard reasoning. This is explained in Criminallian's Aldariaga. And even a Maldesena's original paper. But in this paper, here's a short paper, 2004. They just give the arguments that it works for any single field theory of inflation. But I want to give you a different arguments because it's gonna allow me to introduce the SCFG. So in the last five minutes, I'm gonna tell you all about the SCFG. So that should tell you something about how much we know on the SCFG. So the SCFG, oh, there is some example and actually one of the creators is up there. There is one example by a Ninos Hartman and Strominger. But it involves this Vasiliev theory and it doesn't seem to have a controllable limits in which you look something like Einstein gravity or things we expect inflation to look like. But I think it's fair to say that almost everything we know about the SCFG is coming from some sort of analytic continuation of ADS-CFT. So the statements of DS-CFT, at least the one by Maldesena, is that the wave function of the universe is a function of Gij, that this Gij I've been writing is the beginning of the lecture, can be, this wave function of the universe can be calculated by a partition function of some field theory. So I'm gonna call it CFT, where this Gij is a source for the partition function. So this is the statement. Let me explain what this wave function of the universe is and how it's related to the things we've been computing in the lecture. So the wave function of the universe is the question, if I specify a bunch Davis initial conditions as eta goes to minus infinity, and then I tell, I just wanna keep track of a history that for at late times as eta goes to zero, we'll have the profile Gij of X. So this is a three metric, it's a three-dimensional metric, one dimension less than the bulk. And the question is, what's the amplitude for this process to happen? So this is what the wave function gives me. The actual expectation value, something like Gij is gonna be given by the Born rule. So I have to integrate psi over Gij. Squared times, let's put G of X, G of Y, for example, G of X, G of Y, and then write a psi square, okay? So this is a device that is kind of half step away from the expectation values we've been computing, okay? So, but the interesting thing is that the wave function should resemble a partition function of a conformal field theory. So I'm gonna write the wave function in a way that is very suggestive. So the wave function can be written as the exponential. So let me write it for some Gij, okay? So we know from ADS CFT that the operator that is dual to the metric is the stress tensor. And we are, Zeta and the tensors, they are essentially just components of the metric on the three slides. So we're gonna write Gij, X, Gkl of Y. And if you recall how ADS CFT works, you will not be surprised if I call this, so there's some coefficients here in front of the wave function. Recall that the argument of the wave function is this Gij. And I'm gonna call this thing Eijklxdy16mnijklnzyxzx, plus et cetera, okay? So just from this formula, it's just a way of rephrasing this formula into something a little bit more useful, okay? And finally, I will write down the expression for the Zeta two point function. So suppose I know this coefficients and I'm just, it's just a relabeling. I'm just calling them the correlation function of the stress tensor in some random field theory that looks like a conformal field theory. So this is gonna be given by minus one times two times the real parts of Tt. So this is in momentum space. So you just have to Fourier transform what I wrote here, rewrite it in momentum space. And the Zeta three point function, I'm just doing the Gaussian integral by expending these exponents here.  This is all without delta functions, okay? The real parts are coming from, sorry, it's too real, too real. Real parts are coming because I have to mod square the wave function, okay? Yeah, they're connected Green's functions. It's just some ansatz for the wave function that makes it easy to relate the three point functions to these putative CFG things here. But yeah, it's just a way of writing the wave function that agrees with what you should see from perturbation theory. I'm just calling these coefficients correlation functions of the stress tensor. So because it's Zeta, this is the trace of the stress tensor. So actually in a conformal field theory, all of this is zero, which is just stating the fact that to see the Zeta fluctuation, I need to have a breaking of the series symmetries, okay? Because I have to go to this unitary gauge. But so it's almost conformal field theory because we want to be in quasi-decider space. So I have to break a little bit conformal symmetry in order to generate a trace of the stress tensor. But the interesting thing now that I have a minus couple minutes is that it relates these expectation values to correlation functions of the stress tensor. And at three points, you should be very happy because at three points, we know these things, right? We expect at least in a conformal field theory to know these things. So I'll have a little bit more to say about that tomorrow. But I want to just explain from the point of view of the CFT where these soft theorems come from. So the soft theorems just come from the word identity for the stress tensor. So that's the fact that the stress tensor is conserved inside of an expectation value is all we need. So this is the soft theorem. So this is a story about the background wave and calculating something in the background wave. I have to treat every case separately. I have to treat squeeze limits with a diagonal versus squeeze limits with I meant this type of squeeze limits in which I take the diagonal to zero versus squeeze limits where I take this size to zero. It's all separate and I only get the Mauder Center results. You need to work much harder to get this result. This result is from 2002. This result is from 2010. So it took people a while to figure out this thing. From the point of view of the CFT, all soft limits follow from this formula because when I take a functional derivatives of the covariant derivative here, I'll actually pick some context terms. So it's the fact that the stress tensor inside of a correlation function is conserved modular context terms. So when I translate this statement into a statement for these inflationary correlators, then I'll get all the soft limits. So I think that's kind of neat. So this is, even though the SCFT doesn't exist, this connection kind of already teaches you something, which I find nice. So because I'm five minutes over time, let me stop here.


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Hi everyone, this is Mike Kramer of Mock Capital with a weekly check-in. Today is Monday, July 17th, and it's around 6.30 New York time. So tomorrow we'll be getting retail sales. Estimates are for retail sales to rise by 0.5 percent month over month. That would be higher than last month's reading a 0.3. X sales, X auto up 0.3 percent versus last month 0.1. X auto and gas up 0.3, down from 0.4. Retail sales control group up 0.3 versus last month 0.2. So overall looking for a slightly hotter retail sales in the month of June. When we look at some other data this week, you're going to be getting things like housing starts, continuing claims, which have been a really big number along with initial jobless claims. These numbers have tended to come in below expectations now for some time. So again, a lot of people are watching these numbers closely to get a sense if the labor market is showing any signs of weakening, and then of course we're going to get some of the leading index also on later this week. So plenty of economic data to go around. Also this week we're going to get earnings. Most notably we're going to start with really Wednesday morning Goldman Sachs's reporting results and then later that afternoon you're going to get results out of Tesla and Netflix and then Thursday you're going to see things like Johnson and Johnson and Taiwan Semi, which is I think an important one given that the this is one of the fabricators for NVIDIA. And so you're going to want to see what Taiwan Semi says because that could have a spillover effect into the semiconductor space overall and this recent AI frenzy that's been going on in the market. So you'll want to pay attention to things like that. Also, it's important we just start off with we'll start off with Goldman Sachs because again, Goldman Sachs is on Wednesday but also because Goldman Sachs is one of the largest components in the Dow. So Goldman's results could have some big significance in terms of how the Dow performs. You can see Goldman overall hasn't really performed very well, hasn't really kept pace with the recent equity market rally. So a move higher in Goldman could certainly help the Dow, which has been just basically stagnant here, unable to get through this 34,600 region. It's tested it a number of times going all the way back to November of last year. Now, today, Monday, we got up to that level again, 34,585. We got, you know, briefly above 34,665 today, but just couldn't seem to maintain it. So Goldman could be one of those stocks that really helps to push the Dow over that final hurdle. It's interesting because while you've seen the NASDAQ and the S&P 500 really push to newer highs, the Dow has basically done nothing now for some time. And again, as we've talked about, the Dow has been largely associated with or largely traded with this Australia 200 index, which has been also highly correlated to it. And you can see that both indexes are really not performing very well, just kind of status quo, and you can also overlay the DAX over that as well. And while the DAX has performed better than the Dow overall, both indexes have basically, you know, stalled out, I would say, for the lack of a better word since really around March. So this is sort of an important point for Goldman and for the Dow, because also, you know, you already had UNH report results, and UNH was up 7% on Friday, and it was up another 1% today. And UNH is the biggest weighting in the Dow. And again, so far, haven't been able to break through. You can see also that we're getting very close to this upper bound of resistance on the RSI. So if the Dow is going to break out, it certainly seems like that time needs to be now, because again, you have already have one of the largest components or the largest component report. All of that positivity is already reflected in the price. So you're going to need really Goldman to potentially be the catalyst that gets the Dow over that hump that maybe allows it to climb back up to around 35,400 or 35,500, about 1,000 points higher than where we are right now, which would really put it more in line with the S&P 500, because you can see the S&P 500 right now is knocking on the door of 4,530. And that's been the level that's also been goes back and is the equivalent to where the Dow would be if it were to move higher. The S&P today had a rather strange move higher today. And I say it was strange because we rallied pretty sharply almost all day long. And then in the final basically 30 minutes of the day, we gave all of that, we gave back a big portion of those gains and then managed to also close below the highs that we saw on Friday, which came early Friday morning. And so this could be a number of two things, right? I mean, obviously if the S&P 500 gaps above these highs tomorrow morning, all this negativity that we've seen at least in the last half hour today goes away. And the market could potentially continue on its upward path. And it looks like again with option exploration this Friday, the big gamma level for this OPEX appears to be around this 4,500 level. The next level is at 4,550. So if you take out, if you can take out early tomorrow this 45 and this 45 call it 45, 35 region, there's not much to say that the S&P 500 doesn't rally to 45,50 or maybe even 45,75 in the next couple of days. There could even be some people taking a shot and hoping that they can get this thing up to 4,600. But suddenly if tomorrow's opens down here and we open lower, there's a case that could be said that you start looking for some of these lower gap fills. So I think tomorrow's opening is going to play a critical point in terms of where this index goes and whether or not it tries to push a little bit higher going into options exploration. When you look at the NDX, it also moved higher today. It closed right at the highs though of Friday. So not quite as a negative a close as the S&P, but again like the S&P you really want to see these highs taken out pretty quickly because then after that again as we spoke about last week, there's really no real resistance until about 16,000 on the NASDAQ 100. Also we've been watching the dollar weaken pretty dramatically and this is also helping to add to a little bit of the equity market move higher because when the dollar weakens it helps to ease financial conditions and easing financial conditions can help to push equity prices higher. So the dollar broke through some really key levels here around this 100 to 101 level. You can see today we're trading around 99.10 or so broke some key uptrends in the RSI that was kind of pointing towards a stronger dollar for some time and that reached oversold levels with the index going below 30 and you can even see that the dollar went below the lower Bollinger Bands suggesting that maybe the dollar is a little bit oversold here at these levels maybe we're in for a little bit of a period of a little consolidation so to speak now obviously you get stronger retail sales you could see the dollar lift and maybe retest the breakdown at 101.5 but otherwise I mean for further downside here in the dollar you're looking at 99.15 or so again and if you look at the Euro remember last week we were talking about or maybe two weeks ago we were talking about a potential reversal pattern in the Euro and we were talking about if we got over this 110 region it would invalidate it and that's pretty much what happened right and so when when head and shoulder patterns don't break down and they continue to they can form continuation patterns and you can see that the Euro rose right to resistance here at we'll call it 112 or so in terms of the Euro where it goes from here again like the dollar looks a little bit overbought you can also see that it it also got above its upper Bollinger Band so it seems when you look at the Euro you look at the dollar they're both telling you the DXY they're both sort of suggesting that we might be in a little bit of a period here where we see a little bit of a retracement in both the Euro and a retracement in the dollar index and perhaps the Euro comes back down and tests the 110, 111 region before it makes the next move up towards this 114, 115 area now what's important about the Euro also is if we go back and take a look at it from a longer term perspective and we start drawing in you know trend lines in terms of how the Euro has performed historically you know you can make an argument that the Euro has room really to rise all the way back to 116, 117 before it really hits a long-term downtrend that may pose a threat to it right and and so that suggests that you know again this next level here could be telling because if we get through 112 again it looks like you could be testing a longer term downtrend at around 116 or so so there's a lot of things happening this week I think there's a lot of points in time where you could be seeing some really important moments in the market because again we have option expiration on Friday you also have VIX OPEX on on Wednesday and just really quickly to throw in the VIX OPEX what this is looking like at least is that you have you know a lot of open interest in the calls all up here and these are all due to expire worthless unless we see some sort of move higher in the VIX over the next couple of days that may be a bit of a challenge given that you also have lots of puts which are likely to keep it you know suppressed but you know you can't discount and rule out the fact that there might be a push higher in the VIX in the next couple of days to try to get some of these calls into the money and maybe take some of these puts out of the money and that would certainly that certainly would be a pain trade and that would be a counter to what I think many people might be thinking about but again that's something to think about the VIX did rise today it has been you know up you know three of the last four days the VVIX was also up today and that's been up three days in a row so certainly signs have implied volatility starting to pick up it looks like for now at least the VIX has made a low at this point maybe it makes a lower low but given the OPEX on Wednesday and then on Friday it looks like maybe that may start to reverse course as we go through the rest of the week anyway that's all I have for you today hope you have a great week bye