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Javier Olmedo: Okay, this is our speaker today is by embracing he is going to talk about the fate of ambiguity is in Luke quantum cosmology on primordial our special. Thank you.

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Javier Olmedo: My dog essentially deals with some of the duties and

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Javier Olmedo: Background Hamiltonian constraint and some related issues like our goal in the beginning when we started this project was to understand how some modified loop cosmology lead to

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Javier Olmedo: Variations in the primordial power spectrum and different regimes and in that study we encountered one more ambiguity, which was crucial for us to solve.

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Javier Olmedo: And I would, and on behalf of the authors, I would really like to thank Javier Olmedo for various insightful discussions which led us to overcome some of those obstructions.

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So let me go to Slide one

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Javier Olmedo: So the outline of my talk is that I will give some brief introduction of various ideas, this has something to do with the

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Javier Olmedo: Progress and perturbations in look Congress mythology, a huge amount of work very hard work has been done in the last decade.

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Javier Olmedo: By various groups, I will be just summarizing, some of the things that I would come blend beauties in the background dynamics, which result from different visualizations.

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Javier Olmedo: Of the hundred million constraint, I would label these is MLK you see one or modified Groupon and cosmology one and modify Groupon cosmological

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Not have these details are given in my talk in last February 2019

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Then I will discuss about them using the choice of the momentum of the scale factor or by eight in the Hamiltonian for scale operations.

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This ambiguity has been noted earlier,

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Javier Olmedo: Detailed quantitative study of this ambiguity. So we thought like, while we are addressing the ambiguity in the Hamiltonian constraint, we should also try to see, like, what is a quantitative aspect of this ambiguity.

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Javier Olmedo: Then I will talk about the comparison of the primordial power spectrum and different models which is L QC with with this. The second ambiguity ML QC one and ML QC two

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Javier Olmedo: And the one difference which we have from most of the studies is that we are going to set initial conditions in the contracting right again like they happened a couple of weeks earlier which have also deal with this.

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Javier Olmedo: Initial conditions. The reason why we want to do these initial conditions will become clear, especially for ML QC one and then I will summarize the doc.

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Javier Olmedo: The main goal of the talk is to understand what is the effect of different ambiguity is on the primordial scale a tensor power spectrum in different regimes.

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Javier Olmedo: Not only the regime, which is interesting observations, but we want to ask a question like, if they are different ambiguity is what are the regimes in which

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Javier Olmedo: These ambiguity is leave any trace the main caveat of this talk is that we are going to assume the validity of the dress metric approach as understood in Luke quantum cosmology so far.

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Javier Olmedo: And we are going to assume that is valid for modifiable QC models. So in this talk, I'm going to focus completely on the dress metric approach, but the same study can be done for different approaches also

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Javier Olmedo: Let me go to slide to brief introduction

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Javier Olmedo: Results several others in the Standard Cosmological Model and it gives us a framework to explain the formation of large scale cosmic structure in the universe.

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Javier Olmedo: However Bodega and will include showed in 2003 that inflation is passed incomplete. There is a singularity. In the past, it doesn't resolve the Big Bang singularity.

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Javier Olmedo: And it has been long expected that the quantum gravity affects would resolve this big bank singularity and provide a Planck scale extension of the inflationary paradigm

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Javier Olmedo: In last 20 years we have seen new quantum cosmology results various cosmological singularities and in 2012 even Google IBM taker and William Nelson demonstrated

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Javier Olmedo: That are non singular extension of the conventional inflationary paradigm exists in all cases so that you can extend inflation all back to the Planck scale and have a non singular inflationary paradigm

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Javier Olmedo: It's also useful to know that there are then there are inflationary attractors for various models and new quantum cosmology, or is appropriate models as well as a nice appropriate models.

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Javier Olmedo: This is a very important result because generally a nice appropriate a nice appropriate cause various problems for inflation, but

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Javier Olmedo: We have shown that for a nice appropriate delicacy inflationary attractors exist.

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Javier Olmedo: And there has been a lot of work on understanding the probability of inflation to offer starting from the bombs and I'll be a sticker like screechy Sakura me and David Sloan in particular have shown that the probability for inflation to occur in El que se is extremely large like 99.999%

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Javier Olmedo: A very important result couple to inflation is that is generally it is generally believe that once inflation. Is there any quantum gravity effects on the quantum perturbations would be washed away.

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Javier Olmedo: But there was a very beautiful resolved by one Google narrow sellers and Parker in 2011 which showed that if you have non bunch Davies vacuum states at the onset of inflation.

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Javier Olmedo: Then they can result in stimulated particle creation and this can lead to departures from standard GR predictions and this result forms the backbone of various investigations concerning CM being lukewarm cosmology.

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Javier Olmedo: The idea is that since lukewarm cosmology has a very non trivial pre inflationary dynamics we even though we do not know what is the exact initial state in coming from quantum gravity.

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Javier Olmedo: There can be choice of states which can lead to excitations of the bunch. Davis vacuum or non punch Davis vacuum state.

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Javier Olmedo: And these can potentially provide a very important tool to understand the quantum gravity affects NCLB

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Javier Olmedo: So the main thing is that inflation is very natural and Luke quantum cosmology and this result holds also in modified Groupon and cosmology models.

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Javier Olmedo: There is a mechanism by which we can we, which we can figure out like that. There is our quantum gravity states also quantum gravity states which will be non bunch Davies states and they can lead to stimulated particle creation and lead to departures from

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Javier Olmedo: The non trivial goal is to have an agreement with blank data or CSV data.

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Javier Olmedo: While giving a window small window to test loop on the smallest your new quantum gravity affects this is a very non trivial goal and but this code has been achieved in various, various seminal papers in last 10 years

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Javier Olmedo: Well, let me talk about some of the results. This is a very incomplete list.

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Javier Olmedo: And in the last decade, the cosmological permission to quantum small you have been explored in for mainly for different approaches work has been done by various groups these mainly for different approaches. Other deformed algebra approach.

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Javier Olmedo: Introduced by Martin boy walled Orlean brow grain and others, the dress metric approach introduced by one of Guru be asked taker and William Nelson, the hybrid approach introduced by

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Javier Olmedo: The metric group and Javier on middle and the separate universe approach on which will sell even has been working on.

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Javier Olmedo: And more recent times the dress metric approach and the hybrid approach have gained most attention.

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Javier Olmedo: And the common thing between these two approaches is while there are different assumptions and different approximations in both approaches.

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Javier Olmedo: Quite different. But the common thing between them is that they both utilize the for quantifies perturbations over loop contest background. Generally, the results are qualitatively in agreement between these two approaches and also other approaches.

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Javier Olmedo: So as I was mentioning a very non trivial issue is whether we can have a window so that we can have a

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Javier Olmedo: Some measurements coming from Luke quantum gravity or lukewarm cosmology, while being in agreement with the data is a very non trivial result.

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Javier Olmedo: And this result has been shown in both approaches to exist. The address metric and hybrid approach and some of the very interesting normal results which are

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Javier Olmedo: Which are certainly like the highlights are the following. Like, first of all, there exists some choice of initial states, such that there is a suppression of power for large Angular scales, both in both approaches.

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Javier Olmedo: Yvonne Boris Bollywood and Shane up have worked out very detailed aspects of non wash entities do bounce and, more recently,

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Javier Olmedo: Some are like last line is not coming, but more recently, like the the yesterday there was a paper by our sticker and his colleagues or British school and Sheena.

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Javier Olmedo: In which the showed that there. The, the ongoing tension in the CMT data concerning the power anomaly and the lensing amplitude can be elevated if we use

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Javier Olmedo: It. We use it. We use the states from some of the states from new quantum cosmology. So these are very substantial very substantial results.

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Javier Olmedo: Okay, so let me come to the next slide, which is the dress metric approach. I'll be focusing on the dress metric approach.

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Javier Olmedo: In this in this talk, completely. This approach was introduced by a one acre and William Nelson in 2013

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Javier Olmedo: The main idea is based on understanding quantum field theory on quantum space times, which was introduced by a shaker what recommends again you reclaim indoor ski.

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Javier Olmedo: In 2009. The idea is that the quantum state is saying is a tensor product of the bedroom state sign out and the perturbations states Ivan

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Javier Olmedo: With the background states satisfies the free Hamiltonian evolution equation which is minus is bad Delphi sign out equals x not

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Javier Olmedo: Sign off the exact form of this site x naught is not important, but we can think of it as an evolution operator in in master skill or field loop on the smaller

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Javier Olmedo: The back the production states Simon can be written as a function of beauty and fi Wi Fi is the scale or field which is used as a clock and Q AMP T refer to the

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Javier Olmedo: Scale and tensor perturbations this state Simon is such that the back reaction of the perturbations on the background is negligible, there's an assumption in the best metric approach and this

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Javier Olmedo: Is called as a test field approximation on the dress Madrid approach, a lot of work has been done also numerically to justify this assumption for various states.

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Javier Olmedo: It turns out the dynamics of Psy one is completely equal to that of a state which you always on a metric, which is dressed with some new quantum cosmology corrections.

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Javier Olmedo: And this metric is is this metric G tilda a, a, b, which can be expanded in terms of a Tilda, and the conformal time eater. We are a tilda is computed

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Javier Olmedo: Through these expectation values wear a hat corresponds to the operator corresponding to the scale factor in address metric approach. And similarly, we can find the relationship between the conformance time and define

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Javier Olmedo: All these expectation values are computed using sharply peaked sign out states. So that will be an assumption which will be carrying forward.

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Javier Olmedo: And let me just state, some of the main assumptions in our analysis. So we are going to consider sharply peak states on the effective geometry. So I'm going to assume that such happily peak states exist and and reliable effective geometry for elk UCLA UC Irvine and ML QC to exist.

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Javier Olmedo: And we are going to assume that some important implications for potential important implications from subtle infrared issue. And if you see, which are also pointed out, and

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Javier Olmedo: What X talk recently and have been mentioned in the paper by Kaminski for an Oscar and I haven't asked, they can be successful at rest and in my private communications with

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Javier Olmedo: With a big sticker anyone like I have understood that these can be successfully addressed for sharply peak states and that is assumption in which we are going to work on.

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Javier Olmedo: We are going to assume that they are no back reaction effects. And I should mention that there are studies on incorporating back reactions which might be valuable for this particular studied, both by the Madrid group.

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Javier Olmedo: And also by the by, by Susan Chander and Thomas team. And recently, so I'm going to assume these caveats essentially like I'm not addressing any of the

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Javier Olmedo: Underlying technical assumptions or conceptual issues in the dress metric approach. I'm going to take dress metric approach to be valid for

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Javier Olmedo: Various models. And I'm just going to introduce small ambiguity is on top of them to understand if the dress metric approaches valid, then how do these ambiguities affect the primordial power spectrum.

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Okay, so let me go to the next slide.

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Javier Olmedo: It is useful for the talk to

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Javier Olmedo: Write down these Hamiltonian for scale and tensor perturbations. So this slide essentially does that because this slide is important for that.

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Javier Olmedo: The both address metric approach and the hybrid approach they utilize the seminal work of language in 19 9400 use canonical formulation for cosmological perturbations in the eight using EDM variables.

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Javier Olmedo: I'm going to assume massive scale or field with our potential Wi Fi, even as half empty or five square, it's going to be minimally couple to gravity and especially flat FM W space time

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Javier Olmedo: And following languages for and taking into account all the constraints. It turns out that one can write down

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Javier Olmedo: The perturbations in terms of the physical degrees of freedom switch or one scholar mode and to tensor mode and this killer mode.

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Javier Olmedo: And the tensor modes. These the scale and mortar, essentially, the more kind of captured by the kind of society variable.

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Javier Olmedo: And the Hamiltonian can be written for the scale and mold. This HS in this particular form using the McConnell Sasaki notation us and PS. It's conjugate momentum with a potential omega you scare

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Javier Olmedo: Which has all these stumps various dumps and these terms essentially come from language work similarly, we can write down for the tensor perturbations.

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Javier Olmedo: What is the Hamiltonian for tensor perturbations and the Hamiltonian for those tensor perturbations.

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Javier Olmedo: Doesn't have omega Q scare and there is a simple relation between this Ps and PT we're up to some numerical constants and eight by G and so on.

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Javier Olmedo: So in this slide. It is important to notice that there is a dumb omega Q scare in this perturbation Hamiltonian for the scale of perturbations.

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Javier Olmedo: Which contains this by a scare in the denominator, as well as one over by in this time in the denominator. So, that that is an important thing to notice from from this slide.

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Javier Olmedo: Rest all is very standard. There is nothing new here. So let me go to the next slide. So you can from this Hamiltonian. You can find the equation of motion.

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Javier Olmedo: And that equation of motion, turns out to be this first equation which is essentially the McConnell society equation within cosmic time

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Javier Olmedo: You double K three k dot and a squared plus omega till dusk or by till dusk gear up equal to zero. So if you remove told us this is essentially the classical equation, but in the dress metric approach we essentially recover the same equation, but with

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Javier Olmedo: But with omega tilda square and they told us gear here. The Omega till dusk here is determined from the dress metric approach in the following way.

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Javier Olmedo: We take the expectation values of this omega hat square operator in this particular way. We had this omega had essentially is the operator corresponding to the omega Q which are introduced in the last slide.

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Javier Olmedo: And the terrestrial approximation using sharply peak states the background quantities in the Macondo society equation which is above

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Javier Olmedo: They can be just replaced by their analogs in the effective dynamics. So that is the that is the beauty of of this test feel approximation. Once we have states where the test real approximation is valid.

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Javier Olmedo: We can just start using the effective dynamics to understand what are the effects in the McConnell society equation.

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Javier Olmedo: And once we have solved for this skew case from them. What kind of society equation we can find the primordial power spectrum which is generally computed at the end of inflation.

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Javier Olmedo: With a primordial power spectrum. Is this a cube by two pi scare and more of UK scare buzzy scare with some normalization condition which are not showing with z equals five dot edge and similarly for the for the tensor power spectrum.

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Javier Olmedo: Okay, so let me go to the next slide.

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Javier Olmedo: The. It turns out that the promotional power spectrum which we get from the dress metric approach, as well as the hybrid approach.

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Javier Olmedo: And I think also from the deformed algebra approach is particularly of this is this form, it has essentially three regimes which I'm going to explain

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Javier Olmedo: First of all, let us. Notice that the the quantum quantum the smallest effects define the characteristics scale help you see which is defined as period of a double prime by a were prime reference to the

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Javier Olmedo: Derivative with respect to the confirm our time this scale is set by the underlying quantum geometry essentially by the area gap. And this turns out to be 3.20 in plant units for KL QC

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Javier Olmedo: So all the modes which are which are having like K which is greater than kale, QC, they do not see the underlying quantum gravity background.

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Javier Olmedo: And this is what generally are referred to as the ultra violet regime. So in this primordial power spectrum which is being plotted this regime like this red. Red Line vertical red line denotes a full scale QC and all the modes on the right of it are in the UV regime.

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Javier Olmedo: It also turns out that depending on where the bounce occurs, which is, which gives us the scale of the value of the scale of field at the bounce, there is a minimum observable K

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Javier Olmedo: Which corresponds to the k which is entering the horizon today.

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Javier Olmedo: In our observable universe. And that K observable turns out to be point eight, nine again in LTC when five b is chosen as 1.15

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Javier Olmedo: The reason why five is chosen as 1.15 is to have this nice little window to have some observable signatures of local number smaller

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Javier Olmedo: In the primordial power spectrum. If this five years chosen very large, then these signatures will be washed away.

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Javier Olmedo: If this fiber has chosen to be quite small, like point eight or something, then they will be huge deviations from the, from the CME observations. And the model will be rolled out

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Javier Olmedo: So this is the right window this fiber corresponds to some particular value in this nice little right window in which we have compatibility with observations.

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Javier Olmedo: As well as some signatures that are said large angler large angler skills coming from new quantum cosmology.

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Javier Olmedo: So this k observable defines this line this vertical line and kL QC defines this particular line which is the ultra violet regime.

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Javier Olmedo: Then there is a the what what we should notice that the ultraviolet regime modes essentially correspond to those most which were inside the huddle horizon.

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Javier Olmedo: All throughout in the evolution till the inflation started and as the inflation progresses these more start coming out of the horizon, and since, at that time, we know that then spectrum scale invariant

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Javier Olmedo: These modes are they come out of the horizon. They are frozen and then the lead to a scale invariant spectrum.

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Javier Olmedo: In the middle like these are the most which were essentially coming out from the Hubble radius in the in the super inflationary regime or just after the bounce.

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Javier Olmedo: And coupled with the information which we have for the stimulated particle creation from evens earlier, we're

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Javier Olmedo: Essentially they need to various of solutions here a lot of oscillator behavior is found for this intermediate regime.

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Javier Olmedo: So I'm going to call this as an intermediate regime or oscillator regime in some papers.

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Javier Olmedo: One finds that all the regime left to the UV regime is called infrared regime, I would like to distinguish these two regimes following some of the previous work of brow brain and others.

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Javier Olmedo: Then there is a far infrared regime or the infrared region which is also set by which, in which also like this, the

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Javier Olmedo: Skills the spectrum can be scaling Varian for certain choice of initial states.

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Javier Olmedo: And here we are going to choose the forethought or a diabetic states for which the spectrum. Again, turns out to be scaling variant.

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Javier Olmedo: It is essentially scale invariant, because these are the most which were in the contracting branch and they left the horizon in the Hubble radius in the contracting branch and they are frozen after that.

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Javier Olmedo: And one can show with some competitions that against scaling variances recovered for these infrared regime.

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Javier Olmedo: So essentially, there are three regimes, the infrared regime, the intermediate regime, which is oscillator in nature and the UV regime.

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Javier Olmedo: And there is a window between the scale. You see, and observe you observable and these are the modes of interest to us, which can potentially serve as a tool to test the quantum gravity affects

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Javier Olmedo: So let me come to the next slide.

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Javier Olmedo: So I'm not going. This was my introduction part in which I've introduced that what the what is the basic framework, we are working with. What is the primordial power spectrum. Nothing is new, new here so far. All this is well established and I'm going to move towards the ambiguities.

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Javier Olmedo: Okay, so the first time you do, I'm going to treat is the regularization ambiguity in the Hamiltonian constraint.

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Javier Olmedo: And that this ambiguity essentially stems from the following that in the loop in lukewarm. The smaller

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Javier Olmedo: The Hamiltonian constraint is essentially of this form the Euclidean term and the lorenza and Tom and the Euclidean term and the lower engine term they look like this.

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Javier Olmedo: Generally what we do in Luke quantum cosmology is that before quantization we combine these two terms.

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Javier Olmedo: For the specialty flat model and then we do the standard quantization using called enormous of the hashtag barbero connection.

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Javier Olmedo: It was noted by ending in mind 2009 that we could we could do differently. And we could treat the lorenzen term independently from the Euclidean term as as

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Javier Olmedo: as as as expected to be done in full loop quantum gravity. And if you do that.

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Javier Olmedo: Then you can use identities on similar identities on the classical face face as we generally use in lieu of small g and then express in terms of foreign armies and obtain another Hamiltonian constraint.

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Javier Olmedo: This this result was rediscovered by Andre adapter and cross linger in 2017 from very different construction giving, giving this result a lot of robustness.

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Javier Olmedo: There was another ambiguity, which was mentioned by Yang in mind the paper, which leads to ML QC two in which the idea is that one replaces

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Javier Olmedo: The extensive curvature term in this in the lorenzen piece with the hashtag or barbero connection using the cemeteries of the specialty flat FM W space time and then one mantises as as one would do quantization Lupo number small

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Javier Olmedo: Consult MLC one and ML QC to both lead to very different dynamics.

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Javier Olmedo: Very different equations that a different dynamics can do quantum the smaller but you do certain miracles, like the lukewarm cosmology and ML QC to still have a lot of qualitative similarities

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Javier Olmedo: So let me just show you how the Hamiltonian look like in MLC one the essentially the effective Hamiltonian looks like.

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Javier Olmedo: Looks like in this equation there Vee tells us about the volume of the universe lambda square is proportional to LP square which captures the area gap and B is essentially

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Javier Olmedo: The, the connection over the scared of the trial in the classical in the classical universe, it captures the Hubble rate in the classical universe.

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Javier Olmedo: It turns out that there are two branches which which are which are present for MLC one for be

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Javier Olmedo: And it is very important to carefully switch from one branch to another branch at the bounce. This is very unlike what happens in loop on the smaller ML QC to this is a peculiar feature of mfcc one

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Javier Olmedo: But one can one can switch from one branch to another carefully and then one finds that there is a bounce density, which is

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Javier Olmedo: Which is Rosie, which is the bond sense into quantum cosmology divided by four gamma scare plus one.

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Javier Olmedo: Is the sponsor density smaller than the bounce density in Luke quantum cosmology. So it's Rosie is the bomb standstill Groupon and cosmology.

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Javier Olmedo: Similarly, we can find the effective Hamiltonian for ML QC two which is slightly simpler and but it has similar factors like of science kill MW by to etc. And in this case, the bounce happens at slightly larger

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Javier Olmedo: Bounced density than new quantum cosmology.

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Javier Olmedo: And I'm and this plot, where am taking gamma to be 0.2371 can see the one can see the qualitative difference between three models.

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Javier Olmedo: The red curve corresponds to loop on them. The smaller the green cove corresponds to a milk. You see, to which looks pretty similar.

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Javier Olmedo: But the blue curve corresponds to ML QC one. So, one finds that ML to see one mL QC two and Luke quantum cosmology. They all agree.

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Javier Olmedo: After the bounce in the post bonds regime. They give us very similar qualitative universe, which is large and which has a very small space time curvature

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Javier Olmedo: In the past, even though ML QC one leads to a very rapid expansion which is captured by the digital revolution.

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Javier Olmedo: It turns out that it has a constant. It's a constant curvature space time with the the plank in curvature and the emergent cosmological constant which is also a plank in density

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Javier Olmedo: So I want to see one gives a very different result from ML QC to an LPC in the pre bounce regime.

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Javier Olmedo: Also in the bounce regime there are quality. There are at least like quantitative differences between three of them. It is just, it is just qualitative similarity between them that, then the post pounds, they look similar.

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Javier Olmedo: Some of these effects have been studied in various papers now in last in last two years.

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Javier Olmedo: Some, some things which I should mention that, unlike Groupon cosmology, where the quantum defense equation is of the second order in ML to cnn msnbc to itself. It's of the fourth quarter. It's much more richer equation and

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Javier Olmedo: We understood. It's infrared behavior inconsistency of the infrared behavior recently using one human stability analysis.

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Javier Olmedo: Ml to see one results in an asymmetric bounds and the universe has blank curvature in the rebounds branch with an emergent cosmological constant

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Javier Olmedo: One thing, one should notice that this emergent cosmological constant is only a feature of if you use them improve dynamics.

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Javier Olmedo: If you don't use improved dynamics. And if you use mu naught scheme or old quantization well QC turns out that the emergent matter changes its nature and behaves more like a string gas or has an effective equation state of open model.

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Javier Olmedo: So it depends on how you are regular rising, the areas in loop on the smaller G. Like, it depends. Like, what is it that immediately.

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Javier Olmedo: Affects. What is the emergent matter on the other side ML QC two results in a symmetric bounce with a classical pre bombs regime as an LPC but with a different bounced angsty so it's closest

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Javier Olmedo: To el que si and both the models, I'm able to see what animals. You see, to lead to much more complex Friedman equation which is quite

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Javier Olmedo: quite complex to rights are not writing it, which has higher order corrections than raw score in LTC

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Javier Olmedo: We have found inflationary attractors, for me, Lucas, you want to see to both sold as well as we have found that the probability for inflation to occur is extremely large

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Javier Olmedo: As large as in LTC so all the results about the background dynamics, which were important for inflationary space times in look Congress small g. They have been recovered in last two years for me to see on an MPC to

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Javier Olmedo: Even recently started the effects and primordial scale up our spectrum for me to see one. And finally, like as in Luke quantum cosmology.

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Javier Olmedo: One has a generic resolution of stronger, which are similarities in the effect of space time there is a generic resolution of stronger, which are similarities for both Hamill to see one and ML to see do assuming the validity of the fact of Space Time. Description

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Javier Olmedo: Okay, so now let me come to the

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Javier Olmedo: Once we have introduced these models. How do we set up the initial conditions for different models for the perturbations.

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Javier Olmedo: It turns out that in all of these three models, if the initial conditions of the background are given at the bounds of the bounds of each of these respective models.

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Javier Olmedo: And the only free parameter is firebox one can show that rigorously that the only free parameter is five pounds.

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Javier Olmedo: And as I mentioned that for the existence of the non trivial window to capture quantum gravity affects one should choose the five bonds suspect Gail Qc is greater than k of observing, observing

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Javier Olmedo: Now key observable depends on this is the key observable is a minimum K which is observable today. And this key observable depends on what is underlying five pounds.

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Javier Olmedo: Now for in order for comparison with each of them, since all of them have different background dynamics give slightly different inflationary folds.

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Javier Olmedo: To compare three of them and to also compare our work with previous work in, in, in Luke quantum the smallest one perturbations.

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Javier Olmedo: So what we did was to set up the initial conditions, such that there are always 72 he folding during inflation, which means that irrespective of whether one is dealing with coupon and a small

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Javier Olmedo: Animal to see one or MPC to we are going to require that as soon as the inflation starts

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Javier Olmedo: From the onset of the classical inflation till the end of inflation. They are 70 to 80 folding. So there is an assumption which we made

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Javier Olmedo: This is just to set up the initial conditions, one can do without this assumption easily and none of the results will be affected, but one will have to be just careful how to compare different models in different regimes.

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Javier Olmedo: If you make this assumption for 72 he holdings, which is generally the assumption may which which which is generally what was required in loop on the smaller

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Javier Olmedo: That is why we make this assumption for all the models that it turns out we can set up. What is the five pounds in Luke quantum cosmology. What is a five pounds in MLC one

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Javier Olmedo: And what is a five pounds in ML QC two. So here I've written down. What are those values in LTC the five pounds turns out to be 1.15

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Javier Olmedo: Plank in plank units. He helped you see is determined from the, from the quantum geometric. It has nothing to do with five bonds.

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Javier Olmedo: The background dynamics in LTC determines the scale QC which turns out to be 3.2 but this five bonds equal to 1.15 determines what is a key observable, which is, which turns out to be point eight nine.

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Javier Olmedo: Same exercise enamel to see one. Now if you require 72 he folding should occur during inflation puts five pounds at a higher value as 1.27 in plant units.

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Javier Olmedo: CML to see one is determined from the background dynamics to be 1.6 and a observable turns out to be point six five. In this case, for this five pounds.

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Javier Olmedo: Similarly ML QC two for the sample to a folding to occur during inflation, the five bonds, turns out to be 1.04

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Javier Olmedo: Came across it was determined from the background dynamics and key observable turns out to be 1.11 so each of them have different key.

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Javier Olmedo: The key of the characteristic length because each of them have different background dynamics, each of them have been quantified in a different way. So that is one thing.

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Javier Olmedo: Each of them have different key absorber because of the way we have said the initial conditions that they should have the same number of a folding during inflation.

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Javier Olmedo: Then what we did was we we unlike most of the previous studies. We didn't set the initial conditions at the bounce.

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Javier Olmedo: We wanted to set the initial condition in the contracting branch. One of the reasons was also you want to test

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Javier Olmedo: How do predictions. Do predictions very if you put the conditions in expanding in the contracting branch.

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Javier Olmedo: So what we did was once we have fixed these values from the bounce we evolve the universe backward from the bounce at starting from five pounds.

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Javier Olmedo: And specify the initial conditions for perturbations. As for other aerobatic states which are essential, which has been shown to be essential in the dress metric approach for the correct regularization

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Javier Olmedo: At P equals minus 1.1 hundred 25 and again like this value of t is not nothing canonical about it. We chose different values and tested the results for different values. And this is just one preferred value which we chose

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Javier Olmedo: The so for the LTC and ML QC two, which was the fourth on a diabetic states and in ML QC one. It turns out that since the evolution in the pre boss regime. It makes us a digital revolution.

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Javier Olmedo: We can choose the bunch. Davis vacuum state as the initial state, as was also done by you want to wallow in in in in a recent paper. So for me, Lucy one, which was a bunch Davis vacuum stayed in the contracting branch.

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Javier Olmedo: At some far away from the bounce and then ML to see an ML QC two and look quantum cosmology, the set, we use the photon a diabetic state at t equals minus 1.1 into 25

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Javier Olmedo: But the results which we are going to discuss. They don't depend on the where you set the initial conditions we tested the robustness and the results do not change where you set the initial conditions.

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Javier Olmedo: Okay, so let me now come to the second ambiguity.

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Javier Olmedo: The second ambiguity.

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Javier Olmedo: Essentially deals with what happens to these inverse powers of high in omega squared term. So remember, like, omega squared term.

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Javier Olmedo: Has two terms this omega squared term was coming in the Hamiltonian for the skill of motivations. It has two times one is one hour pious year term. And one is one hour by

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Javier Olmedo: Now this is a subtle issue. So I'm going to spend a little time on this. So classically pi, which is the momentum of the scale factor.

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Javier Olmedo: It can be written in this particular form we from the Hamilton's equations for the scale. Scale factor A dot one can easily find this equation.

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Javier Olmedo: And even lively uses this equation in Luke on the smaller than it is problematic because at the bounce aid or become zero. Similarly animals to see why animals to see to

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Javier Olmedo: A dot become zero and Fi is going to be zero, which means one of our pie and whenever pious scared. We are all going to blow up. So this is not the right way to do it. So strategy was introduced

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Javier Olmedo: By eBay Yvonne and Willie Nelson in the 2013 paper. And this is just one of the strategies. There are various strategies to handle this. So I'm just pointing out the strategy.

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Javier Olmedo: What they did was they considered the vanishing of the background constraint classical background constraint which is which is this equation. Essentially, this comes from the DM

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Javier Olmedo: Variables. This immediately implies that one hour pious care if one, one can divide this by the volume AQ this term gives us the energy density of the internet on field.

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Javier Olmedo: So there is infrared on field energy density row. So, one can ride on one hour pious care from this equation in this particular form.

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Javier Olmedo: And after some competition, one finds an expression analog to omega Q scare, which I'm going to call omega plus care for, for the reason which will become clear. Soon, which looks this which looks like this particular form that FS scared of quantify 24 5G by row times five dot

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Javier Olmedo: Once you have this form, then one can use a scale factor and the energy density from effective dynamics.

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Javier Olmedo: And then one can plug in into this omega plus

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Javier Olmedo: Square and then one can solve them kind of society equation and one can find the primordial power spectrum. So this was one of the strategies.

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Javier Olmedo: It turns out that this expression is only true if you are starting from the bounce and then looking forward, and we were looking at expanding branch.

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Javier Olmedo: Again, this is one subtle issue which has been noted earlier by different people in different groups.

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Javier Olmedo: And it turns out that you're starting from the contracting branch this plus sign should be actually the minus sign.

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Javier Olmedo: So if you're starting from the contracting branch, which is what we are starting from

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Javier Olmedo: This omega one will not have omega plus gear but domain or minus square with a minus sign here, and then at the bounce this has to be changed to omega plus here.

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Javier Olmedo: There is another strategy and then other strategies that instead of doing this way one can just solve the effective dynamics and obtain the solution for pie.

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Javier Olmedo: And then login that from that pie obtain omega scare and then study the implications for that.

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Javier Olmedo: In the power spectrum. And this was briefly studied in recent paper on non wash aunties by one Morrison Shana, and they are they are they are they are

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Javier Olmedo: The result, essentially, was that there are differences, but they are only significant and then for regime, but that was done only for el que si

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Javier Olmedo: And as far as I think starting from the bounce regime, not from the contracting branch.

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Javier Olmedo: Okay, so let me come to the next point in this subtlety. So as I was mentioning that in the contracting branch.

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Javier Olmedo: The Omega Q school has a slightly different form. It has a minus sign in sort of a plus sign in the second term.

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Javier Olmedo: And it turns out that the plus sign and this minus sign, cause this discontinuity at the bounce, which is depicted in this figure. So the black of corresponds to in this figure in omega minus care which is coming from the contracting branch and the omega plus here corresponds to the

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Javier Olmedo: This potential in the expanding branch.

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Javier Olmedo: So, this is this, there is a discontinuity, that it's small discontinuity. But what we noticed is that if you do not get take care of this discontinuity.

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Javier Olmedo: Then there can be some serious implications, at least for me to see one that is what we noticed. So what we did was our first attempt was to artificially smoothing it out.

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Javier Olmedo: And from in a very small regime like starting from this is this t is in the plank. Second, so even in the half of plank second before and after the bounce.

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Javier Olmedo: We just artificially smooth and out with a smoothing function, which has given which which is this cost lambda be this course lambda be quickly becomes unity.

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Javier Olmedo: On both sides and just disappears within, within a fraction of a second. So it has a very small trace and we checked. Some robustness test.

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Javier Olmedo: With with this this moving function and we didn't see anything subs anything significant at all from from this movement.

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Javier Olmedo: I should mention that this moving function is is right now was partially inspired by the strategy by one of the tools which was used in the hybrid approach. So it was inspired from there.

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Javier Olmedo: And let me come to this point that make them to the second ambiguity. The second strategy, which is used, which was used in the literature for this omega square. So, so far, like what I have talked about is that there is a

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Javier Olmedo: Ambiguity in dealing with by a there was a conventional approach and respected approach and in order to take from the contracting branch of the expanding branch.

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Javier Olmedo: And to remove this discontinuity at the bonds. We had to introduce this small smoothing function artificially which was inspired by an independent strategy used in the hybrid approach. So let me come to work process. What is a strategy which is generally used in the hybrid approach.

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Javier Olmedo: So in this strategy. The idea is

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Javier Olmedo: Not to use the classical Hamiltonian constraint at any level but replace one hour pious care by this particular town.

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Javier Olmedo: In the, what kind of society equation wherever it appears

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Javier Olmedo: And the interesting thing is that if you do this replacement for the classical background Hamiltonian constraints. So suppose you start from the idiom constraint for the background.

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Javier Olmedo: You introduce the system, then what one gets is essentially the QC effective Hamiltonian constraint. So this is like a shortcut method to go from the classical Hamiltonian constraint to the LTC constrained by using this particular replacement

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Javier Olmedo: Similarly, one can do the for the one over by a term so it cannot be obtained just from the square root of this one hour by your spare time, unfortunately.

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Javier Olmedo: The reason is, again, tied to the discontinuity and in the hybrid approach. This is done cleverly

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Javier Olmedo: By taking one hour pie and not as scared of this, but something very close, which looks like this.

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Javier Olmedo: Which has again that AWS lambda be built in so that costs lambda be which I introduced in the smoothing function.

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Javier Olmedo: Essentially is already present in this one hour pie and there is no need to reduce any further smoothing

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Javier Olmedo: It turns out that once you have this course lambda be which quickly becomes one within fractions of blank second on both sides of the bonds.

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Javier Olmedo: Then it captures right sign of pie and there is no discontinuities. So you can start from the contracting branch and you can go through the bounce and go into the expanding branch.

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Javier Olmedo: So we label this with these replacements, or one hour pious care and one hour pie in the Omega Q scare in the fact two dynamics.

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Javier Olmedo: As omega square effective. So I'm going to call the the previous approach, which has a smoothing function as just omega square plane omega square and this one is omega square effective

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Javier Olmedo: Because essentially, like if you do this replacement in the classical background Hamiltonian constraint, you get the effective Hamiltonian of help you see

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Javier Olmedo: The same replacements, the similar replacement, not the same. I'm sorry. Similar replacements can be done for me to see one animal QC to buy carefully studying

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Javier Olmedo: The behavior of the hammer effective background Hamiltonian, looking at the discontinuity and then choosing the right one of our pie. So, so the similar construction can be done for that. So I will, I will tell what the results are for that.

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Javier Olmedo: Okay, so let us first come to a interesting point, which is the effect for

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Javier Olmedo: Effect of omega square versus omega square effective and why it is non trivial to understand this.

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Javier Olmedo: So I showed you the McConnell suck equation in in cosmic time but we can rewrite that equation in conformal time in this particular way. We are new.

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Javier Olmedo: Is a scale factor times the Macondo Sasaki variable. Q And this equation has this case care.

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Javier Olmedo: And an s wer S is omega not scare minus A double prime by a omega not scare is just a label here to label either omega square or omega squared effective whichever choice, you want to make

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Javier Olmedo: A double prime by a essentially captures the Richie scholar or a double six A double prime by a HQ essentially captures which is killer. So this is

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Javier Olmedo: This is many times, called as the curvature town in literature. So I'm going to use the same notation here. So here we see two plots.

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Javier Olmedo: The first the plot on the left hand side corresponds to t from zero going backwards in the pre bonds regime and the plot on the right hand side goes from zero

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Javier Olmedo: Were to to forward for in time for the expanding branch, the bounce happens or equal to zero in both the figures. And we can see here

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Javier Olmedo: That in both the figures that the curvature term in the post pounds branch is very large, except this regime where it crosses

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Javier Olmedo: Where it becomes smaller than omega three and Omega squared, effective and after that this term against rises and again becomes larger.

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Javier Olmedo: Than a double then both omega scares circulate times this term, the curvature term is hundred times larger than any of the omega squared terms. So that is the first thing you notice

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Javier Olmedo: Also in the post pounds. This term is very large compared to both the terms, it is true, we can see that omega square effective term which is obtained from

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Javier Olmedo: Using the one of the tools in the hybrid approach is tend to the part six times greater than omega square. Add the bounce the bounce this blue curve is standards are six times greater.

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Javier Olmedo: Than omega square at the bounce.

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Javier Olmedo: Quite large compared to this at exactly at the bounce, which I'm not saying t equals zero here, so it at the bounce. This is standards are six times larger.

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Javier Olmedo: But still, both of these terms are much smaller than the curvature. So what we find is that in the post box branch one Kanye only thing

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Javier Olmedo: That omega not scared. Tom in this s doesn't contribute compared to the curvature. So on my name. Only thing that there are no effects of omega square vs omega square effective

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Javier Olmedo: In the pre in the post transplant in the preborn smart branch, the situation is similar. Even though omega square effective is very large compared to omega square, but still the curvature time is very large is always larger compared to both of these potential terms.

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Javier Olmedo: So one might nicely thing that since the storm is always smaller than the curvature term, we might not have

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Javier Olmedo: Any relevance for the power spectrum and these will not leave any signature.

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Javier Olmedo: But first of all, it is noticeable that there is a big change in big change in the amplitude of the omega square and omega square of effective omega squared omega square effect, especially near the bounce, there is a there is a big difference between them.

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Javier Olmedo: One can find similar conclusions for me to see what animal QC to by looking at what is their omega square and what is their omega square effective and plotting the curvature term. A double prime by

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Javier Olmedo: So nicely, one might conclude that there is probably going to be no no no relevance of this ambiguity in power spectrum.

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Javier Olmedo: But let me show you now some results for the primordial scale up our spectrum. First, the skill of our spectrum and Luke quantum cosmology.

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Javier Olmedo: So this is just for loop quantum cosmology and in the left figure I have plotted the rat triangles and and the and the blue circles.

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Javier Olmedo: For the average power spectrum. And this is taken in for the wave numbers going from 510 to the power minus six to 50 so it covers the entire range from the far infrared to ultraviolet modes, the average is taken by taking small wins and which which have five we have numbers each

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Javier Olmedo: On the right, I've plotted the relative difference between the power spectrum computed from omega square and the power spectrum converted from omega square effective

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Javier Olmedo: And this relative differences defined like this to have the models of Pac Man SPS prime divided by the models of ps plus PS prime

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Javier Olmedo: So what we are comparing these two figures is the skill power spectrum and look on them. The smaller, there is no mo QC here right now.

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Javier Olmedo: And we are looking at omega square and omega square effective and here I've also mentioned k star which is a star is a pivot mode in plank units, which is, which turns out to be 7.28

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Javier Olmedo: And the five pounds is is 1.15 in this particular model. So, case studies essentially used to permit tries the amplitude of the power spectrum. So it turns out to be 7.28

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Javier Olmedo: In this particular in this particular case. So what we find is that there is a huge difference. More than hundred percent difference

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Javier Olmedo: In a small interval which is 10 to the power k by k star which is going from 10 to the minus four and the power minus three. This is essentially the

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Javier Olmedo: Corresponds to the boundary of the intermediate regime and the infrared region, which I was mentioning earlier.

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Javier Olmedo: But as you go towards the ultra violet regime. The difference between difference coming from this ambiguity between the choice of omega square arming Oscar effective becomes very, very small. It becomes less than

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Javier Olmedo: Point 1%. So these are, this is the regime, which is crucial for us of observational so most of the observable modes are lying in this region and for these observable modes.

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Javier Olmedo: That this ambiguity gives a difference of only point 1% so everything is safe. Everything is robust, as well as, as far as we see from Groupon and cosmology.

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Javier Olmedo: At least for the linear perturbation in the scale of power spectrum. Now, this increases to about very large difference as we go to the intermediate scales.

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Javier Olmedo: And in the far infrared regime again like this difference is is quite is quite substantial, it can go up to 10% for for various

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Javier Olmedo: For various values in the infrared regime. So the main conclusion from this slide is that the dough. There is a big difference in the intermediate raging because of this.

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Javier Olmedo: ambiguity in the choice of momentum of the scale of momentum on the scale factor in the ultra violet modes which correspond a central to the altar.

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Javier Olmedo: To the observable modes. There is there is very, very little difference, but the traces there in the infrared region as well as Intermediate Region.

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Javier Olmedo: So let me come to the next slide, which is the primordial skill or power spectrum in ML to see one.

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Javier Olmedo: So same exercise, not on enamel QC one. And again, we see that the

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Javier Olmedo: That we get the scale invariant spectrum and ultraviolet regime and also the scale invariant spectrum. And then for regime, but with a very large amplitude

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Javier Olmedo: Very noticeably very, very large amplitude compared to what was the situation and look on the smaller this very large amplitude is essentially coming from very large emergent cosmological constant and the pre bonds regime, and then this the case that changes the pivot mode.

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Javier Olmedo: Changes to 5.57 essentially again determined by the five pounds and and which is determined by the background dynamics.

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Javier Olmedo: For omega squared effective here, we use the following substitution, one of our pious care and one of our pie. You went by this particular form this one hour pious care if you put this one over pious care in the EDM background constraint.

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Javier Olmedo: Then you will recover the effective Hamiltonian for me to see one. So same strategy like what was what was done for LTC

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Javier Olmedo: This one of our pie has a slightly more complicated form which, instead of just a simple cosine lambda has this one minus two squared plus one science lambda be

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Javier Olmedo: This term comes from this very careful matching between the two branches which I was mentioning at the bounce. So between those two to do that. Careful matching

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Javier Olmedo: One has to include this particular term and some are like this one, and that is not showing, but the same time comes. If you take the omega square

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Javier Olmedo: Which which which which was the first me first choice and ambiguity is and instead of that cosine lambda be

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Javier Olmedo: You place the, you have to put the same time if you do not put the same time, then there is no, then there is a discontinuity between them. So this provides a smooth matching between them.

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Javier Olmedo: So what happens for the error in this and let us see, like, what is the main result for me to see one. And these results were already there for for

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Javier Olmedo: In anyone's paper he he was the first one to obtain this results for the scale up our spectrum.

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Javier Olmedo: What we have done in addition to what we did earlier was to look at this ambiguity about by in more in in in detail we introduced this ambiguity and we looked at the quantitative details.

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Javier Olmedo: So as one form. We also confirm his, his result independently, that it was a large amplification in power in the IRA gene compared to help you see

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Javier Olmedo: And as I will show this is also in comparison to ML QC two. And as I was mentioning this is due to the Planck scale emergent cosmological constant in the pre bonds epoch.

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Javier Olmedo: There is a very fast growth for amplitude in the intermediate regime. So let me show you this slide again like this is a very fast growth compared to this growth which we see in the Intermediate Region.

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Javier Olmedo: This is a characteristic of the MLC one, essentially, then this plot, I have plotted the in the plot here at plotted the relative difference for them to do to the power spectrum.

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Javier Olmedo: For the choices of omega square versus omega square effective and we see like there is a 10% error or 10% relative difference. And then for the gym.

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Javier Olmedo: And sometimes greater than even hundred percent of the relative difference in the intermediate regime and it varies significantly

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Javier Olmedo: And then remediate regime. But as we go to the ultraviolet regime, this difference becomes very small this difference actually becomes less than point 1% and the UV region.

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Javier Olmedo: So the same result similar result as an LPC when you go to the UV regime, the ambiguity of pie and how you choose by really doesn't matter. This is not really surprising because

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Javier Olmedo: The other at modes, where those which never felt the background dynamics all till when they reach inflation.

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Javier Olmedo: And this the background dynamics has felt by the modes which were in the intermediate reaching those were the ones which are coming out of the horrible horizon in the super inflationary phase.

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Javier Olmedo: And during the bounce. So these are the ones which are going to see the most of the most of the relative difference. And here we are not surprised that the relative difference is very, very small in the UK region.

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Javier Olmedo: So let me come to ML QC two

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Javier Olmedo: In MLC to we for the first choice of omega scare the choices similar instead of cosine lambda be one has to choose cosine lambda be by to

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Javier Olmedo: That is that is essentially a reflection of that you don't have a second order quantum difference equation. Now you have a fourth order quantum difference equation and you have to choose this angle carefully. So, this is this this choice just essentially comes from the Hamiltonian constraint.

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Javier Olmedo: Then to obtain omega square effective we do this following substitutions. Again, guided by what was done in rupal number small g for omega squared effective

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Javier Olmedo: If you plug in this one hour pious care in the EDM constraint, you will get them all to see one constraint, I will go see to constraint.

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Javier Olmedo: And for me for one of our pie. We take the square root of this put a minus sign and then plug in the same factor here cosine of lambda be to again like this is done to have the proper matching in the in the pre bonds and the proposed bonds regime.

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Javier Olmedo: So here we see that there is a relative difference in the IRA gene which is larger than 10% it again shoots up 200% at the boundary of

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Javier Olmedo: Intermediate and the IRA, which we can see here there's, there is a significant difference between the black and red triangles.

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Javier Olmedo: The for the which gives you the power spectrum for omega squared, effective and omega square. And again, here the case star.

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Javier Olmedo: Changes because the background dynamics has changed and five be has changed in this as we go to

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Javier Olmedo: As we go to the ultraviolet regime. Then again, this ambiguity leaves very little signature and the error or the relative difference. I'm sorry, is less than point 1%

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Javier Olmedo: So it's about 30% in I urging the highest entire gym for compared to execution ML QC to it's about hundred percent and the boundary, like in all the all the other two models and slowly decreases and then becomes

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Javier Olmedo: Less than point 1% in the utility. So all three models have one same characteristic feature that if you go to the UV regime, which is, which contributes most to the observable modes.

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Javier Olmedo: The ambiguity essentially leaves very little trace. But if you are interested in the intermediate regimes or then fire regimes, for some reason, if you're interested in the contracting branch, then there is a significant trace of this ambiguity.

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Javier Olmedo: OK, so now let me compare the scale up our spectrum for accuracy and ML QC two

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Javier Olmedo: So we are not comparing the scale of our spectrum for QC animal QC one or ML QC one animal QC to because I'm able to see one is very, very different power spectrum.

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Javier Olmedo: Like they there is no comparison between them. That's qualitatively very different. So the El que se nemelka see to power spectrums can be

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Javier Olmedo: Compared rigorously. So here in the left trader. The spectrums are computed, you're just using omega square for both of them.

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Javier Olmedo: And for the right figure they are computed with omega square effective for both of them. So we see some differences, whether you compute using omega square and omega square effective

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Javier Olmedo: I'm not showing the relative difference block or the relative difference between el que si and ML QC two is larger in the IRA gene for if you choose omega scare

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Javier Olmedo: You can see that there is a larger difference in the IRA. If you choose omega square and this different stones are to be greater than 60%

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Javier Olmedo: Omega squared effective also gives a large difference, but it is less than 40% so omega squared gives you more difference than omega square factor.

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Javier Olmedo: There is a significant difference at the boundary of the IR and intermediate regime. So this is the boundary of the infrared and the intermediate regime and we can see there is a significant difference between this group points. I'll to see and the ML QC two points.

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Javier Olmedo: If you are doing omega scare factor this differences slightly smaller if you're doing omega square

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Javier Olmedo: And the relative difference essentially varies from one 200% like it's a huge variation going from hundred percent to almost 1% in the whole intermediate regime.

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Javier Olmedo: And if you go to the ultra violet regime, this difference turns out to be less than one person. So if you if if you compare two different regularization

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Javier Olmedo: With the same choice of ambiguity for how you treat the momentum of the scale factor, then the differences are larger

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Javier Olmedo: Those differences are always they always turned off, it turned out to be larger than if you had just one model and you were

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Javier Olmedo: choosing different ambiguities. So the difference between the models is larger than the difference between the ambiguity and just one model. So that is the conclusion for the scale or power spectrum. You can do the same exercise for the 10th power spectrum. And here we compared for

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Javier Olmedo: In the left and the right figure I have the relative difference just for advocacy and ML QC two. I'm not voting for MLC one because that is a huge difference compared to the difference of advocacy and ML QC two

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Javier Olmedo: So first of all, we noticed that the amplitude and ML QC one is extremely large in the IRA gene very, very large. And I mean, it has a very sharp growth going as I think I scared to the power minus three. In this regime in this intermediate regime.

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Javier Olmedo: On the other hand, the elk. You see, and ML QC to power spectrum, it's, it's quite similar in all the regimes, but there is still that there are still relative difference in particular, like in the IRA gene, the relative differences greater than 50%

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Javier Olmedo: In in the intermediate regime, it varies widely from more than 130% to about 1% at some point. And as you go to the UV regime, then this difference becomes quite small, less than

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Javier Olmedo: Less than 1%

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Javier Olmedo: I'm sorry this should be UV region nada Richie me that's a typo. Okay, so this one is essentially the, the, the details of the various primordial power spectrum. So let me just summarize

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Javier Olmedo: The main results. So different treatments of Laurens in terminal Hamiltonian constraint in in once we are trying to contest lukewarm cosmology. It is already non trivial changes in the physics of bounce and also have the pre bounce.

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Javier Olmedo: But still magically ML QC two beers qualitative similarity with LTC they are qualitatively close to each other, even though the equations are

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Javier Olmedo: vastly different. For both of them physics is close to each other. However, ML QC one results in an asymmetric bounce with the prevalence University, the plank in curvature

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Javier Olmedo: The. That was the first ambiguity. The second ambiguity came from the treatment of by a which is the momentum of the scale factor in the scale or perturbation Hamiltonian

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Javier Olmedo: This has been performed in rotation in different ways we use the approach, which was widely used so far interest metric, as well as one which was

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Javier Olmedo: Inspired from the hybrid approach. So I have to say, like, I didn't use the hybrid approach at all except one of the tools which they which they use one one small tool.

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Javier Olmedo: Which they used to handle it. So this work had nothing to do with hybrid motivations. But still, it can be done similar things can be done with in full glory of the hybrid approach.

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Javier Olmedo: The results, the main result is and this confirms and generalize is the result by one

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Javier Olmedo: The results between LTC ML to see one and ML QC to they agree in the UV regime. This is the regime, which contributes most to the miserable modes.

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Javier Olmedo: with exception of the small window. I was talking about between kale QC and cubes variables.

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Javier Olmedo: And this regime, the ambiguity of pie is also a little significant because you are far away from this corresponds to the motor strong see the background dynamics of the bounce.

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Javier Olmedo: And once you are far away from the bounds. Once you an inflationary regime. There is no difference between different choices of omega scale because the universe has become quite classical and both ambiguities result in the same omega square

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Javier Olmedo: So main result is that the qualitative production predictions for the observable modes for the linear perturbations in the dress metric approach our robust

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Javier Olmedo: To the considered Hamiltonian regularization and beauties, and the pie ambiguity. That's the main is also the dose prediction. The desk metric approach robust

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Javier Olmedo: Well, let me continue, but yes. The question is, like, if I have different organizations and if I have these ambiguities do the lever trace the answer is yes, they leave a significant trace

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Javier Olmedo: But in the regimes of infrared modes and the intermediate regimes, where they were oscillation. So these were the most which were which came out of the horizon in the contracting branch.

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Javier Olmedo: And the moles, which are which came out of the horizon in the super inflationary face. So these are the two regimes in which

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Javier Olmedo: These ambiguity is really make all them. Be good is really to leave their signature significant signature.

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Javier Olmedo: The relative difference between a to c and ML QC to can be as large as 50 200% in the IRA gene for any one choice of omega scare you take the relative differences 50 200% in the ideology.

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Javier Olmedo: And the also an intermediate regime, this difference can be very large 200% depending on what choice of omega square or watch is of by you take

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Javier Olmedo: Ml QC one is very different from both of these models like in the IR implemented regime that we use qualitatively different signatures with very large amplitude and its rapid growth of amplitude both force killer as well so

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Javier Olmedo: If you are just focusing on one particular model like QC or MLC to our ML to see one than the different sources of high end result in at least 10% relative difference in Iowa.

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Javier Olmedo: And intermediate regimes with some regions of intermediate regimes small windows of intermediate regimes even giving you 100% relative difference

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Javier Olmedo: The key question is, even though there are these traces in the intermediate regime, which tend to die out towards the UV regime.

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Javier Olmedo: Are there any signatures, which are possible. If we go to for example like non kosher entities. If you go to non gosh entities, then the role of intermediate. The modern intermediate regime become significant

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Javier Olmedo: As far as the work of Yvonne Boris and sure enough show. So it is quite possible that these ambiguity is might leave some distinct signatures observable signatures when we studied on it. So that is a future work which which we will do pretty soon. Thank you very much.

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Javier Olmedo: What's yours.

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Francesca @ Western: Maybe I have a question, a very basic one is. FRANCESCA Hi, but um

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Francesca @ Western: Can you comment on how your choice of the number of before the that is a very reasonable one, but in America is how these

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Francesca @ Western: And influence your results.

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Javier Olmedo: So first of all, the choice of the holdings hold influence the main results here. So again, like they will be regimes different regimes IRA Jim intermediate regimen UV regime.

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Javier Olmedo: And the results will be essentially the same. What will changes, whether you are pushing the these differences more in the intermediate regime. Suppose you take

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Javier Olmedo: Initial conditions which give you extremely large holdings like hundred 85 holdings, then there is no window left to observe any of the

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Javier Olmedo: Of the Loop Quantum cosmology or underlying quantum geometry effects. So all these relative differences will move more and more towards the infrared scale and move towards the

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Javier Olmedo: Infrared skills and Intermediate Region. So qualitatively, they will be no change these results will be seen

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Javier Olmedo: But it will become if the, if your goal is, OK, I take hundred e folders and I want to still observe observable signatures.

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Javier Olmedo: Potentially, I'm saying just potentially just speculation in on gosh Janet is then it will become much harder even observed them so that is only change which will occur.

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Thank you.

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Javier Olmedo: In 700 plus I have seen that there is

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Javier Olmedo: Like a sharp peak in the power spectrum. That is what in the infrared remote support is not going to be your survival.

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Javier Olmedo: But they would like to know if how to check if this is a trial actually an America ratified, or if it is physical and if it is physical the, you know, any explanation or or in or this enhancement of our

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Javier Olmedo: So probably you're talking more than the peak at the peak maybe let me just come to that slide, you're probably talking about this kind of appeal. Right. So we have checked this for what we did was to check it for different values of five pounds.

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Javier Olmedo: This power spectrum, not only for the five bonds, which I mentioned we also checked for different

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Javier Olmedo: Values of the time at which we said the initial conditions for perturbations, even at the very close to the bounce starting from and pushing it back to very earlier times

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Javier Olmedo: And we have checked for two independent codes. So we didn't find any change and we don't think it's a numerical artifact.

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Javier Olmedo: We do not fully understand why there is this amplification and why there is a difference in amplification compared to El que si and ML QC to, for example,

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Javier Olmedo: So we do not, we do not understand that. But this seems to be a robust effect, and as far as I think this is also effect, which was seen if you take take different types of initial states.

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Javier Olmedo: Like second order a diabetic states, but we do not know the main physical reason why this, why this jump occurs.

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simone: Hi, Brian.

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simone: Hi, Tim. Want to hear your take on the way to go forward with these ambiguities. Would you seek for more input from the full theory in order to reduce them or would you seek for phenomenology inputs to select a viable models.

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Javier Olmedo: That's a very good question. Simone. So one, there are various ways to look at this result. So as I was. So let me just go to that, like the main result as you look at it.

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Javier Olmedo: Is that the qualitative quantitative predictions of at least in the dress metric linear perturbations are very robust to these ambiguities.

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Javier Olmedo: Another way to look at this result is that, well, there are these ambiguities and these ambiguity is are living a signature, but that is signature is very difficult to observe. So I think the

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Javier Olmedo: First of all, it shows that even very simple choices. So these were just starting from Luke quantum cosmology and

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Javier Olmedo: Doing slightly different regularization, so we are still very, very far away from Luke quantum gravity. So even in lieu of quantum cosmology that different choices.

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Javier Olmedo: Which lead to such me duties which can leave some signature in promoted power spectrum. This of course they are. They are certainly not in the observable window, at least for the linear perturbations.

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Javier Olmedo: So it tells that if there are more ambiguity is like, let us say like the kind of work which alecky was doing

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Javier Olmedo: Or which clause and Andrea and Thomas had been doing been trying to generalize these these models, then these ambiguities would be there.

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Javier Olmedo: We have to live with them. And this work only shows that yes, there is a they will leave some praise. Now it depends on which regime. They are leaving. So I think like

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Javier Olmedo: They will be traces remaining. So I think we have to do twofold things. First of all, we have to narrow down these ambiguity as much as possible.

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Javier Olmedo: By seeking a connection with the full theory. That's the first thing we have to do, because unless we do not do that. There is going to be a Degeneres in the results. Because right now, whatever are the results.

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Javier Olmedo: From el que si for the observable modes. Essentially, they all go through for both MLC one and ML QC two and probably for many other models which are still not discovered

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Javier Olmedo: Because all of these results essentially capture the modes which are not feeling the bounce physics of bounds. So they will all go through. So that is a there is a degeneracy there.

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Javier Olmedo: So we have to narrow down, theoretically, what this ambiguity should be. Secondly, we have to go beyond this work, which I have presented so far golden on gosh entities, for example.

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Javier Olmedo: Using very beautiful work. Which one should we not and Boris have done using that as a platform going forward with these ambiguities and trying to find the distinct signatures.

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Javier Olmedo: In the higher order in perturbations and then probably we can rule out some of these models or we can rule out well.

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Javier Olmedo: We can say probably that there are certain favorable situations for some are some models. And there are some very unfavorable models. So, for example, ML QC one

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Javier Olmedo: Already gives very high amplitude at the infrared mode which which probably is questioning that may be like there is an issue with even stability of motivations or not. So there is already some danger there, but I think like we still need to do more work.

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Javier Olmedo: From an observational side and from this phenomenology side to rule out some of the choices and certainly we have to do a lot of work from the full theory side.

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Javier Olmedo: To bring the right cosmological sector to look condom cosmology.

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Javier Olmedo: My personal belief is that if we do the cost if we do the proper derivation of motivations for coming from full theory. Some of these ambiguous won't even exist. They will be one right choice which will be one right natural choice.

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Javier Olmedo: And some of these ambiguous. I was mentioning, or the spy, they should not exist at all.

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Francesca @ Western: But I'm a sensor, someone asked you the question about the traditional way. The, the food theory. I have a question, a very general question about these two and

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Francesca @ Western: I just wanted to have a sense of how you feel about that because so in your analysis, you're considering they would have violated the regime and you were introducing these k koji. A QC and you were considering modes that

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Francesca @ Western: Shorter than this k, you see. So these modes. If you think about the food theory, they should not be there somehow. And there is a tension within the treatment in cosmology, maybe they need

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Francesca @ Western: In an inflationary model to have them. So I would, I would like to hear your comments about this. Where, where are we now, or what is the sense in the community about that.

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Javier Olmedo: But I'm sorry, like I'm confused. Like why these most will be forbidden by the full theory.

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Francesca @ Western: Maybe I have a miss understood that I understood that, maybe, maybe it's my confusion. Then I understood that the

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Francesca @ Western: modes that correspond that that is smaller than K QC they are smaller than the

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Okay, so

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Javier Olmedo: Yeah, so the calc let us talk about accuracy and kale Qc is just sat by the underlying quantum geometry that is Gail QC just scared of a double prime by a

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Javier Olmedo: A double prime mistaken in conformal time. Okay. So given your background dynamics kale QC turns out to be 3.20 or 3.21 in plank units so

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Javier Olmedo: The key observable means that what if you choose any five pounds. If you choose some initial conditions from the background dynamics.

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Javier Olmedo: Which so suppose gives you a five pounds which was taken as 1.15. What is the mode, what is the smallest wave number which is observable today.

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Javier Olmedo: That is what gives herbal means. So today, in the present day Park. What is the mode which is entering the horizon today.

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Javier Olmedo: I can trace back using the background evolution back to the bounce and I can find out what is the key observable minimum caves herbal at at the bounds.

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Javier Olmedo: Because I know the entire background evolutions. It has nothing to do like a observable will be there.

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Javier Olmedo: It, what can happen is that if you do not choose your five pounds carefully. If you choose your five pounds, very large.

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Javier Olmedo: Then you're able to see will become smaller than K observable and then you will have no scope to see the quantum gravity affects if you choose your five pounds too small, then your kale QC will be much, much larger than caves.

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Javier Olmedo: And then your predictions with the blank or web app will be completely ruined from Luke quantum cosmology.

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Javier Olmedo: You have to choose this window carefully such that the five pounds is in a reasonable range.

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Javier Olmedo: Such that KL Qc is just larger than cubes with a small window says that there is an agreement with the data as far as a small window to capture

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Javier Olmedo: quantum gravity affects at large Angular scales. So I don't think it has a connection with Luke quantum gravity is going to rule out key absorber.

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Javier Olmedo: Completely if there is a result which is our well cosmological sector from Luke quantum gravity is such that KL to see is always greater than

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Javier Olmedo: The observable. That will be a very interesting result then probably we won't see anything of quantum gravity in these observations at least using these assumptions.

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Francesca @ Western: Okay, thank you. This clarifies

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Javier Olmedo: Are there any further questions.

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Javier Olmedo: Okay, no less than the speaker again.