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I haven't been able to find a reference to the first who showed that the small density fluctuations in the cosmic microwave background led to the formation of large scale structure as we know it.

I'm not an expert in this, but I'll give it a try.

I think you've got the argument the other way round here. This was never discovered. It was assumed. But why?

- We know for the CMB being a redshifted blackbody-spectrum. This means the CMB consists of photons that were emitted in interaction with matter.
- The CMB fluctuations inform us about the depths of gravity wells that those photons had to escape from (amongst other things). Thus we get a picture of the baryonic and dark matter density at the time of photon-matter decoupling.
- This allows us to set up linear perturbation theory + simulations to look what happens to a uniform gas with the measured fluctuations on top. That's the rough idea.

You could now of course ask "who was the first to implement this?". The earliest theoretical work that I'm aware of (Peebles 1965) linking "a possible universal blackbody radiation" with the formation of galaxies came in the year of the discovery of background radiation by Penzias & Wilson 1965.

## Cosmology's Only Big Problems Are Manufactured Misunderstandings

This large, fuzzy-looking galaxy is so diffuse that astronomers call it a “see-through” galaxy . [+] because they can clearly see distant galaxies behind it. The ghostly object, catalogued as NGC 1052-DF2, doesn’t have a noticeable central region, or even spiral arms and a disk, typical features of a spiral galaxy. But it doesn’t look like an elliptical galaxy, either, as its velocity dispersion is all wrong. Even its globular clusters are oddballs: they are twice as large as typical stellar groupings seen in other galaxies. All of these oddities pale in comparison to the weirdest aspect of this galaxy: NGC 1052-DF2 is very controversial because of its apparent lack of dark matter. This could solve an enormous cosmic puzzle.

NASA, ESA, and P. van Dokkum (Yale University)

If you keep up with the latest science news, you're probably familiar with a large number of controversies concerning the nature of the Universe itself. Dark matter, thought to outweigh normal atomic matter by a 5-to-1 ratio, could be unnecessary, and replaced by a modification to our law of gravity. Dark energy, making up two-thirds of the Universe, is responsible for the accelerated expansion of space, but the expansion rate itself isn't even agreed upon. And cosmic inflation has recently been derided by some as unscientific, as some of its detractors claim it can predict anything, and therefore predicts nothing.

If you add them all together, as philosopher Bjørn Ekeberg did in his recent piece for Scientific American, you might think cosmology was in crisis. But if you're a scrupulous scientist, exactly the opposite is true. Here's why.

If you look farther and farther away, you also look farther and farther into the past. The earlier . [+] you go, the hotter and denser, as well as less-evolved, the Universe turns out to be. The earliest signals can even, potentially, tell us about what happened prior to the moments of the hot Big Bang.

NASA / STScI / A. Feild (STScI)

Science is more than just a collection of facts, although it certainly relies on the full suite of data and information we've collected about the natural world. Science is also a process, where the prevailing theories and frameworks are confronted with as many novel tests as possible, seeking to either validate or refute the consequential predictions of our most successful ideas.

This is where the frontiers of science lie: at the edges of the validity of our leading theories. We make predictions, we go out and test them experimentally and observationally, and then we constrain, revise, or extend our ideas to accommodate whatever new information we obtained. The ultimate dream of many is to revolutionize the way we conceive of our world, and to replace our current theories with something even more successful and profound.

Long before the data from BOOMERanG came back, the measurement of the spectrum of the CMB, from . [+] COBE, demonstrated that the leftover glow from the Big Bang was a perfect blackbody. One potential alternative explanation was that of reflected starlight, as the quasi-steady-state model predicted, but the difference in spectral intensity between what was predicted and observed showed that this alternative could not explain what was seen.

E. Siegel / Beyond The Galaxy

But it's not such an easy task to reproduce the successes of our leading scientific theories, much less to go beyond their present limitations. People who are enamored with ideas that conflict with robust observations have had notoriously difficult times letting go of their preferred conclusions. This has been a recurring theme throughout the history of science, and includes:

- Fred Hoyle refusing to accept the Big Bang for nearly 40 years after the discovery of the Cosmic Microwave Background,
- Halton Arp insisting that quasars are not distant objects, despite decades of data demonstrating that their redshifts are not quantized,
- Hannes Alfven and his later followers insisting that gravitation does not dominate the Universe on large scales, and that plasmas determine the large-scale structure of the Universe, even after countless observations have refuted the idea.

Although science itself may be unbiased, scientists are not. We can fall prey to the same cognitive biases that anyone else can. Once we choose our preferred conclusions, we frequently fool ourselves through the fallacious practice of motivated reasoning.

Schematic diagram of the Universe's history, highlighting reionization. Before stars or galaxies . [+] formed, the Universe was full of light-blocking, neutral atoms. While most of the Universe doesn't become reionized until 550 million years afterwards, with the first major waves happening at around 250 million years, a few fortunate stars may form just 50-to-100 million years after the Big Bang, and with the right tools, we may reveal the earliest galaxies.

S. G. Djorgovski et al., Caltech Digital Media Center

It's where the famous aphorism that "physics advances one funeral at a time" first came from. This notion was originally put forth by Max Planck with the following statement:

A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.

The big problem that many non-scientists (and even some scientists) will never realize is this: you can always contort your theoretical ideas to force them to be viable, and consistent with what's been observed. That's why the key, for any theory, is to make robust predictions ahead of time: before the critical observation or measurement is performed. This way, you can be certain you're testing your theory, rather than tinkering with parameters after-the-fact.

According to the tired light hypothesis, the number of photons-per-second we receive from each . [+] object drops proportional to the square of its distance, while the number of objects we see increases as the square of the distance. Objects should be redder, but should emit a constant number of photons-per-second as a function of distance. In an expanding universe, however, we receive fewer photons-per-second as time goes on because they have to travel greater distances as the Universe expands, and the energy is also reduced by the redshift. Even factoring in galaxy evolution results in a changing surface brightness that's fainter at great distances, consistent with what we see.

Wikimedia Commons user Stigmatella aurantiaca

As it turns out, this is exactly how we wound up with the leading cosmological model we have today, in pretty much every regard.

The notion of the expanding Universe was theoretically predicted by Alexander Friedmann in 1922, when he derived what I have called the most important equation in the Universe. The observations of Vesto Slipher, Edwin Hubble and Milton Humason confirmed this only a few years later, leading to the modern notion of the expanding Universe.

According to the original observations of Penzias and Wilson, the galactic plane emitted some . [+] astrophysical sources of radiation (center), while a near-perfect, uniform background of radiation existed above and below that plane. The temperature and spectrum of this radiation has now been measured, and the agreement with the Big Bang's predictions are extraordinary.

Many competing explanations for the Universe's origin then emerged, with the Big Bang having four explicit cornerstones:

- the expanding Universe,
- the predicted abundances of the light elements, created during the hot, dense, early stage of the Big Bang,
- a leftover glow of photons just a few degrees above absolute zero,
- and the formation of large-scale structure, with structures which must evolve with distance.

All four of these have now been observed, with the latter three occurring after the Big Bang was first proposed. In particular, the discovery of the leftover glow of photons in the mid-1960s was the tipping point. As no other framework can account for these four observations, there are now no viable alternatives to the Big Bang.

The fluctuations in the CMB, the formation and correlations between large-scale structure, and . [+] modern observations of gravitational lensing, among many others, all point towards the same picture: an accelerating Universe, containing and full of dark matter and dark energy. Alternatives that offer differing observable predictions must be considered as well, but compared with the full suite of observational evidence out there.

Chris Blake and Sam Moorfield

With an expanding, cooling Universe that began from a hot, dense, matter-and-radiation-filled state, all governed by the Einstein's General Relativity, there are a number of possibilities for how the Universe could have unfolded, but it's not an infinite number. There are relationships between what's in the Universe and how its expansion rate evolves, and that tremendously constrains what's possible.

This is the only statement that is unequivocally correct in Ekeberg's piece.

Once you accept the Big Bang and a Universe governed by General Relativity, there is an enormous suite of evidence that points to the existence of dark matter and dark energy. This is not a new suite, either, but one that's been mounting since the 1970s. Dark energy's main competitor fell away some 15 years ago, leaving only a Universe with dark matter and dark energy as a viable cosmology to explain the full suite of evidence.

Constraints on dark energy from three independent sources: supernovae, the CMB and BAO (which are a . [+] feature in the Universe's large-scale structure.) Note that even without supernovae, we'd need dark energy, and that only 1/6th of the matter found can be normal matter the rest must be dark matter.

Supernova Cosmology Project, Amanullah, et al., Ap.J. (2010)

That's the key that's so often overlooked: you have to examine the full suite of evidence in evaluating the success or failure of your theory or framework. Sure, you can always find individual observations that pose a difficulty for your theory to explain, but that doesn't mean you can just replace it with something that does successfully explain that one observation.

You have to account for everything, plus the new observation, plus new phenomena that have not yet been observed.

This is the problem with every alternative. Every alternative to the expanding Universe, to the Big Bang, to dark matter, dark energy, or inflation, all fail to even account for whatever's been already observed, much less the rest of it. That's why practically every working scientist considers these proposed alternatives to be mere sandboxing, rather than a serious challenge to the mainstream consensus.

The Carina dwarf galaxy, very similar in size, star distribution, and morphology to the Draco dwarf . [+] galaxy, exhibits a very different gravitational profile from Draco. This can be cleanly explained with dark matter if it can be heated up by star formation, but not by modified gravity.

There are indeed galaxies out there without dark matter, but this is predicted by theory. In fact, nearly a decade ago, a prominent contrarian noted the lack of galaxies without dark matter and claimed it falsified the dark matter model. When these galaxies without dark matter were discovered, that same scientist immediately claimed they were consistent with modified gravity. But only dark matter explains the full suite of evidence concerning the Universe.

There is, indeed, a discrepancy between two different sets of groups trying to measure the expansion rate of the Universe. The difference is 9%, and could represent a fundamental error in one group's technique. More excitingly, it could be a sign that dark energy or some other aspect of the Universe is more complex than our naive assumptions. But dark energy is still necessary either way the only "crisis" is aritificially manufactured.

A plot of the apparent expansion rate (y-axis) vs. distance (x-axis) is consistent with a Universe . [+] that expanded faster in the past, but where distant galaxies are accelerating in their recession today. This is a modern version of, extending thousands of times farther than, Hubble's original work. Note the fact that the points do not form a straight line, indicating the expansion rate's change over time. The fact that the Universe follows the curve it does is indicative of the presence, and late-time dominance, of dark energy.

Ned Wright, based on the latest data from Betoule et al. (2014)

Finally, there's cosmic inflation, the phase of the Universe that occurred prior to the hot Big Bang, setting up the initial conditions our Universe was born with. Although it's often derided by many, inflation was never intended to be the ultimate, final answer, but rather as a framework to solve puzzles that the Big Bang cannot explain and to make new predictions describing the early Universe.

- successfully reproduces all the predictions of the hot Big Bang,
- solves the horizon, flatness, and monopole puzzles that plagued the non-inflationary Big Bang,
- and made six novel predictions that were distinct from the old-style Big Bang's, with at least four of them now confirmed.

The quantum fluctuations that occur during inflation get stretched across the Universe, and when . [+] inflation ends, they become density fluctuations. This leads, over time, to the large-scale structure in the Universe today, as well as the fluctuations in temperature observed in the CMB. These new predictions are essential for demonstrating the validity of a fine-tuning mechanism.

E. Siegel, with images derived from ESA/Planck and the DoE/NASA/ NSF interagency task force on CMB research

To say that cosmology has some interesting puzzles is compelling to say it has big problems is not something that most cosmologists would agree with. Ekeberg discusses the inflationary Big Bang with dark matter and dark energy as follows:

This well-known story is usually taken as a self-evident scientific fact, despite the relative lack of empirical evidence—and despite a steady crop of discrepancies arising with observations of the distant universe.

To argue that there's a lack of empirical evidence for this completely misunderstands what science is or how science works, in general and specifically in this particular field, where data is abundant and high in quality. To point to "a steady crop of discrepancies" is a disingenuous — and I daresay deliberate — misreading of the evidence, used by Ekeberg to push forth a solipsistic, philosophically empty, anti-science agenda.

Many nearby galaxies, including all the galaxies of the local group (mostly clustered at the extreme . [+] left), display a relationship between their mass and velocity dispersion that indicates the presence of dark matter. NGC 1052-DF2 is the first known galaxy that appears to be made of normal matter alone.

Danieli et al. (2019), arXiv:1901.03711

We should always be aware of the limitations of and assumptions inherent to any scientific hypothesis we put forth. Every theory has a range of established validity, and a range where we extend our predictions past the known frontiers. A theory is only as good as the verifiable predictions it can make pushing to new observational or experimental territory is where we must look if we ever hope to supersede our present understanding.

But we mustn't forget or throw out the existing successes of General Relativity, the expanding Universe, the Big Bang, dark matter, dark energy, or inflation. Going beyond our current theories includes — as a mandatory requirement — encompassing and reproducing their triumphs. Until a robust alternative can reach that threshold, all pronouncements of "big problems" with the prevailing paradigm should be treated for what they are: ideologically-driven diatribes without the requisite scientific merit to back them up.

## Who discovered the relation between CMB fluctuations and large-scale structure formation? - Astronomy

Department for Innovation in Biological, Agro-Food and Forest Systems (DIBAF), University of Tuscia, Viterbo, Italy

Copyright © 2014 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

Received 27 March 2014 revised 22 April 2014 accepted 12 May 2014

This is a review of the status of the universe as described by the standard cosmological model combined with the inflationary paradigm. Their key features and predictions, consistent with the WMAP (Wilkinson Microwave Anisotropies Probe) and Planck Probe 2013 results, provide a significant mechanism to generate the primordial gravitational waves and the density perturbations which grow over time, and later become the large-scale structure of the universe—from the quantum fluctuations in the early era to the structure observed 13.7 billion later, our epoch. In the single field slow-roll paradigm, the primordial quantum fluctuations in the inflaton field itself translate into the curvature and density perturbations which grow over time via gravitational instability. High density regions continuously attract more matter from the surrounding space, the high density regions become more and more dense in time while depleting the low density regions. At late times the highest density regions peaks collapse into the large structure of the universe, whose gravitational instability effects are observed in the clustering features of galaxies in the sky. Thus, the origin of all structure in the universe probably comes from an early era where the universe was filled with a scalar field and nothing else.

**Keywords:**Large-Scale Structure, Cosmic Inflation, CMB, Non-Gaussianity

In the past, at the onset of its history, the early universe was hot and dense, a plasma of nuclei, electrons and photons whose mean free path for Thomson scattering was very short. In this universe there is no classical space-time, there is the impassable curvature and density singularity which emerges in the general relativity when the scale factor a approaches zero. A crucial point the zero, a point of infinite density, where what become before the big bang is yet unknown.

10 − 44 s after the big bang, there are two unique parameters: Planck’s mass and length

,.

At these scales, where the Planck density is the continuum tears and the space-time itself ends, all known physics comes to halt, physical observables associated with both matter and geometry diverge and quantum gravity becomes necessary. At such very high density its effects become dominant, and then the predictions of the general relativity, based on the space-time as a smooth continuum, are inapplicable in a regime where space and time may be discrete, and quantum effects dominate.

As the universe expanded and cooled down at the recombination era temperature of

3000˚K, the primordial plasma coalesced into atoms, the photons begin their travel through the universe for 13.7 billion years and their wavelength stretched at the scales of the observable universe. Today, the radiation is observed as cosmic microwave background (CMB), and its measurement, together with the distribution of galaxies, distances to type Ia supernova explosions and others, have revealed that the universe, whose spatial curvature is found to be negligible, is filled with photon, baryons (4%), dark matter (23%) and dark energy (73%).

In our universe, the large-scale structure formation via gravitational instability demands the preexistence of small fluctuations on large physical scales, as galaxies scales

10 24 cm, which left the Hubble horizon in the radiation and matter dominated eras. However, since at these scales there are no causal mechanisms to generate fluctuations, the generation of primordial small perturbations, at scales smaller than the horizon, and their Gaussianity or non-Gaussianity, are crucial questions for the large classes of cosmological models.

At the present, the six parameters Lambda Cold Dark Matter (ΛCDM) model—the simplest of the Standard Cosmological Models [1] -[6] —is the main stone, but the small fluctuations have to be put in by hand, whereas in the inflationary scenario [7] -[31] there are primordial energy density perturbations correlated to the quantum fluctuations of the inflaton field which, once the universe became matter dominated (z

3200), were amplified by gravity and grew into the large structures of our universe [29] .

The existence of these primeval inhomogeneities has been confirmed by the Cosmic Background Explorer (COBE) discovery of the cosmic microwave background (CMB) temperature anisotropies which trace back to the inflation lasting different time intervals in different regions of the universe. Inflation then provides a mechanism to generate, not only the density perturbations which later grow into the large-scale structures, but also gravitational waves.

Today, the inflationary potential had become an indispensable building block of the Standard cosmological theory and can be also considered as part of an extension of the Standard Model (SM) of particle physics that is supposed to describe the fundamental interactions at the level of field theory. If the combination these models will survive the current and future cosmological observations, is likely to be the one chosen by Nature. Thus, the origin of all structure in the universe probably comes from an early era where the universe was filled with a homogeneous scalar field -the inflaton field—and nothing else.

In this early era, the potential dominated the energy density of the universe decreasing slowly with time as the field rolled slowly down its potential slope. The inflaton field perturbation has practically zero mass and negligible interaction. The Fourier components of the primordial density perturbation are uncorrelated and have random phases, and the primeval perturbation is Gaussian. The spectrum of the spatial curvature, defined as the expectation value of at the epoch of horizon exit, defines all of its stochastic properties. The shape of the spectrum is defined by the spectral index given as [23]

. (1.1)

In some models of inflation, is almost constant on cosmological scales, and the curvature and the gravitational spectra, whose features provide the possibility to be observed, are defined as

. (1.2)

In the slow-roll inflation the spectrum is slowly varying, corresponding to a spectral index The primordial spectrum is also slowly varying and the gravitational wave amplitude is predicted to be GaussianIn this review we try to explain how the primeval inhomogeneity have been generated in the initial moments of the early universe, where a heuristic quantum field—the inflaton—was the source of negative pressure and accelerated expansion. In this short inflation stage tiny quantum fluctuations in the inflaton field translated in the density perturbations which, as seeds, grew into of the large-scale structure observed today. In other terms, the quantum fluctuations of the inflaton field were excited during inflation and stretched to cosmological scales. At the same time-since the inflaton fluctuations coupled to the metric perturbations via Einstein’s equations -ripples on the metric were also excited and stretched to cosmological scales. Thus, in the cosmic inflation, since perturbations in the inflaton field imply perturbations of the energy-momentum tensor, and the perturbations of this tensor imply perturbations in the metric, both inflaton field and metric perturbations, tightly coupled to each other, lead to the formation of the large-scale structure.

This review, in which many technical details have been suppressed or simplified, is organized as follows: in Section 2, we revise the key features of the standard cosmological model, the ekpyrotic/cyclic models, the slow-roll inflation model with simple polynomial potentials, the loop quantum cosmology (LQC), which deeply modifies the Einstein equations, replaces the singularity with a quantum bounce and extends the inflationary scenario all the way to the Planck regime. In Section 3, we show how the primordial quantum fluctuations and density perturbations grow via gravitational instability to become the large-scale structure of the universe. In Section 4, we relate the key features and predictions of the large classes of models to the current observations—the WMAP and Planck datasets. In Section 5, we reassume the status of large-scale formation via primordial quantum fluctuations and gravitational instability, revise some of the unsolved fundamental questions, as the origin of the inflaton field, and finally conclude that today, the deepest mysteries of our universe is yet the puzzle of whence it came.

2.1. A Partial List of Available Models

In the development of the inflationary scenario, there are many interesting models: the axion in inflationary cosmology [32] , the hybrid inflation [33] , the eternal self-reprodution chaotic inflationary universe [34] , the inflationary multiverse [29] [35] , the chaotic inflation in supergravity [36] [37] , the string and brane inflation [38] - [41] and others. But, the key features of these heuristic models are still under debate.

The alternatives to the inflationary scenario: pre-big bang [42] [43] , textures and cosmic structures [44] , string gas scenario [45] -[47] , bounce in quantum cosmology [48] , let unsolved one, or more, problems of the standard cosmological model. The same ekpyrotic/cyclic scenario [49] -[51] , introduced as a radical alternative to the standard inflationary cosmology, has its own problems and is yet under debate.

Anyway, the inflationary paradigm is not the only to provide a mechanism for the generation of cosmological perturbations. Other mechanisms have been proposed: the warm scenario [52] [53] , where dissipative effects provide the radiation production occurring in the inflationary expansion stage, the curvaton mechanism [54] -[56] to generate an initially adiabatic perturbation deep in the radiation era, the D-cceleration [57] , an unconventional mechanism for slow roll inflation, the Ghost inflation [58] , where the ghost condensate is a physical field with physical fluctuations, the ekpyrotic/cyclic scenario [49] -[51] , where the field runs back and forth the interbrane potential from some positive value to-∞ and back and the big bang singularity disappears in the endless sequence of epochs, and others.

In the same inflationary scenario there are more than hundred inflation models, where many models provide a mechanism to generate Gaussian and non-Gaussian fingerprints in the early universe whose nature is analyzed in [5] [7] [59] . If the primordial fluctuations are Gaussian-distributed, they are characterized by their power spectrum or, equivalently, by their two-point correlation function. The non-Gaussianity (NG) of the primeval fluctuations is captured by the 3-point correlation function, or its Fourier counterpart, the bispectrum. Different NG configurations (Equilateral, Local, Folded, Orthogonal) are linked to different mechanisms for the generation of non-Gaussian perturbations at different scales.

Here, we just mention few models with detectable amplitude of non-Gaussianity.

Equilateral NG: the single field inflation with a non-canonical kinetic term [60] [61] , the k-inflation [60] [62] , the Dirac-Born-Infeld inflation (DBI) [63] [64] , the general higher-derivative interactions of the inflaton field as ghost inflation [58] .

[There are two types of DBI inflation models. In the UV model the inflaton slides down the potential from the UV side of the warped space to the IR end. This results in a power law inflation when the scale of the potential is high enough. In the IR model, the inflaton is originally trapped in the IR region through some sort of phase transition and then rolls out from the IR to UV side. The resulting inflation is exponential and the potential scale is flexible.]

Folded or flattened NG: the single field with non Bunch-Davies vacuum (NBD) [60] [65] , effective field theories models [66] . Orthogonal NG: Non-Gaussianity in single field inflation [67] Local NG: multi-field inflation [68] .

The comparison between the current and future cosmological observation and the predictions and key parameters of these large classes models will decide their fate. Today, the Wilkinson Microwave Anisotropies Probe (WMAP) [69] -[71] and Planck 2013 [72] -[74] datasets have already ruled out some models and strongly constrained others.

2.2. The Standard Cosmological Models

The six parameters ΛCDM model describes successfully many features of the evolution of the universe—the Hubble expansion, the existence of a CMB with a blackbody spectrum, the primordial D, 3 He and 7 Li abundance, the sum of the masses of the three families of neutrinos, the dark energy equation of state parameter, the number of effective relativistic species, the big bang nucleosynthesis. But, the small fluctuations—which left the Hubble horizon in the radiation and matter dominated eras—have to be put in by hand. Moreover, the model requires that the energy density of the universe has to be tuned near the critical density with an accuracy of 10 , and postulates the homogeneity and isotropy of the early universe, which must extend to scale beyond the causal horizon at the Planck time.

The standard model then, requires unnatural initial conditions at the big bang, is limited to those epochs where the universe is cool enough to be described by physical processes well established and let unsolved several fundamental problems: the homogeneity, isotropy and flatness of the universe, the origin of irregularities leading to the formation of galaxies and galaxy structures, the primordial monopole gravitino and initial singularity, the about 60 order of magnitude between the Planck’s length (10 cm) and mass (10 g), and the actual size (10 28 cm) and mass (10 55 g) of the universe, the vacuum energy problem, which trace back to the idea that the constant scalar field appearing in unified theories of elementary particles could play the role of a vacuum state with energy density in cosmology.

2.3. The Ekpyrotic-Cyclic Universe Scenario

In the ekpyrotic/cyclic scenario [49] -[51] , there are no initial conditions, a bouncing universe replaces inflation, the singularity disappears in the endless sequence of epochs and non-Gaussianities of the local type are produced. In [49] [50] , the scalar field runs back and forth the interbrane potential from some positive value to-∞ and back. In each cycle there is a sequence of kinetic energy, radiation matter and dark energy dominated phases of evolution that agree with the standard big bang cosmology, but the models are not free of conjectures and the key features of the brane world physics are still an open question.

At the present, the cyclic universe [49] [50] faces the question of the metastability of the Higgs vacuum, suggested by the recent measurement at the LHC. (The discovery of a Higgs-like particle with mass 125 - 126 Gev, combined with measurements of the top quark mass, implies that the electroweak Higgs vacuum may be metastable and only maintained by an energy barrier of height h (10 10-12 Gev) 4 that is well below the Planck density [75] ). The Higgs metastability makes problematic for the big bang to end one cycle, bounce, and begins the next. However, on using an appropriate Weyl-invariant version of the standard model coupled to gravity to track the Higgs evolution in a regularly bouncing cosmology, has been found that exists a band of solutions which solve the problem. The Higgs field escapes from the metastable phase during each big crunch, pass through the bang into an expanding phase, and returns to the metastable vacuum, cycle after cycle. Further, due to the effect of the Higgs, the infinitely cycling universe is geodesically complete, in contrast to inflation where a metastable Higgs makes inflation more improbable [75] .

The old [10] and new inflation [13] -[16] have introduced a significant innovation in the Standard Cosmological theory, but in their frameworks there are postulates which are somewhat artificial. The universe, assumed as relatively homogeneous and large enough to survive until the start of the inflation, was in a state of thermal equilibrium from the very beginning the inflation was an intermediate stage of its evolution. The introduction of the chaotic inflation [17] [18] , which describe the evolution of an universe filled with a chaotically distributed scalar field resolved all problems of the old and new inflation. In this model, inflation begins in absence of thermal equilibrium and may occur, not only in the models with simple polynomial potentials as but also in any model where the potential has a sufficiently flat region, which allows the existence of the slow-roll regime. In the simplest model of the chaotic inflation with potential the value of the scalar field determines the existence of different regimes [17] [18] [31] . At energy density of the field there is no classical space-time, and then an universe which emerges from the singularity, or from nothing, has to be in a state with Planck density which can be described as a classical domain.

In this classical space-time domain, the initial sum of the kinetic energy, gradient energy, and potential energy densities cannot be greater than the Planck density

, (2.1)

and the expected typical initial conditions are [31]

(2.2)

In this context, the onset of inflation occurs at the natural condition and its continuation requires within the Planck time.

In the inflation stage, the scalar field runs its potential from to its minimum value. At high potential energy density, the quantum fluctuations of the scalar field are large and may lead to an eternal process of self-reproduction of the inflationary universe. At lower values of the inflaton field slowly rolls down its potential and its fluctuations are small. Finally, near the minimum of the field rapidly oscillates, loses its energy creating pairs of elementary particles, and the universe becomes hot.

In the cosmic inflation with potential, the universe evolution and the inflaton dynamics—in the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric—are governed by the Einstein-Friedmann equations [28]

(2.3)

(2.4)

, (2.5)

where for an open, flat or closed universe respectively, and the term acts as a friction term which slows down the motion of the field In an universe governed by these equations, an inflation stage requires a negative pressure, that is (Equation (1.4)) as in the de Sitter stage.

For a homogeneous universe the above simplifies as

, (2.6)

where

Therefore, for initially large values, the Hubble parameter was also large—which implies that the friction term was very large—and the field was moving very slow. At the onset of inflation, the energy density of the scalar field remained almost constant and the expansion of the universe was very fast. Soon after the onset of this regime the conditions were satisfied, and then the universe was governed by a simplified system of equations

. (2.7)

In this inflationary regime, if the field changes slowly, the size of the universe grew exponentially, , and inflation ended when. Solution of these equations states that after a long period of inflation the universe initially filled with the field grows exponentially as [28] .

In this inflationary paradigm a sufficient slow-roll period is achieved only if the scalar field is in a region where the potential is sufficiently flat. The flatness condition on the potential is conveniently parametrized in terms of two slow-roll parameters built from the derivatives of the potential V with respect to

. (2.8)

Therefore, to achieve a successful period of inflation these slow-roll parameters must be much smaller than one, The parameter can be also written as thus it quantifies the rate of the Hubble parameter H during inflation.

In summary, in the simplest inflation with potential, the realistic value of the mass m is about (in Planck units), the total duration of inflation is

10 − 30 seconds after the Planck time, and in this very short period the total amount of inflation achieved from the onset of inflation at is of the order. Inflation ends when the scalar field begins to oscillate near the minimum of loses its energy by creating pairs of elementary particles which interact between them and come to a state of thermal equilibrium— the matter creation and reheating stages. From this time on, the universe can be described by the standard cosmological model. An universe whose is many order of magnitude greater than the part of the universe which is today seen, cm [31] .

The inflationary paradigm solves many problems of the Standard cosmological models and predicts the existence of inflationary perturbations which serve as seeds which grow into a large scale structure. Therefore, since the detailed features of these perturbations have been observed in the CMB, the predictions of the inflationary scenario appear in agreement with observations However, despite its successes, the inflationary paradigm is conceptually incomplete in several respects. The initial conditions are postulated [76] , the Trans-Planckian issues are ignored [77] , the inflationary space-time is past incomplete, and then inherit the big bang singularity [78] , even though the inflaton violates the standard energy conditions of the singularity theorems [79] .

2.6. Loop Quantum Cosmology

In loop quantum cosmology (LQC) framework [80] -[90] , based on the key features of loop quantum gravity (LQG) [91] -[98] and its underlying Riemannian quantum geometry, space-time and perturbations are quantum, the quantum fields corresponding to the perturbations propagate on a quantum space and the non-perturbative quantum corrections dominate the evolution in the Planck regime, replace the big bang singularity with a quantum bounce and disappear in the low energy regime insuring agreement with the classical general relativity. Thus, the LQC quantum corrections solve the singularity and the ultraviolet-infrared tension: the short distance limitations of classical general.

The LQC novel results are encoded in the modified Einstein-Friedmann equation [87] [100]

(2.9)

where and are respectively the Hubble rate and the maximum energy density. This equation for, that is away from the Planck regime, reduces to the classical Friedmann equation

, (2.10)

where for matter density positive, cannot vanishes, so that every solution represents a contracting, or an expanding universe, whereas in LQC, where the modifications in the Planck regime are drastic, vanishes at the quantum bounce, to its past the solution represents a contracting universe with and to its future, an expanding universe with.

In LQC, the bounce is an effect of the quantized geometry. The quantum effects act as an effective quantum repulsion, allow the universe to bounce back from a collapse, and then resolve the singularity without fine tuning initial conditions.

From the modified Friedmann equation, and the standard conservation law, it follows that also the deceleration equation is modified to

. (2.11)

This result implies that, since can be positive, inflaction (in presence of an appropriate scalar field) is generic in LQC and does not require unlikely initial conditions.

Moreover, the LQC pre-inflationary dynamics of natural conditions at the bounce [101] -[104] , allows to include quantum gravity regime in the standard inflationary scenario, to extend the evolution all the way from deep Planck regime, to ignore the back reaction also in the pre-inflationary epoch, to solve the trans-Planckian problem and to account for the inhomogeneities seen in the CMB, which are the origin of the large scale structure. In the LQC application, for the LFWR space-time with quadratic potential [100] [104] , the LQC predictions in the low energy regime agree with those of the standard inflationary paradigm: both are compatible with the power spectrum and the spectral index reported in the seven years WMAP data.

Thus, in the low energy regime, LQC and standard inflationary paradigm predictions, compared with the cosmological observation, have the same fate. But, their predictions for the state at the onset of inflation are different: in the standard inflation scenario the onset state is the Bunch-Davies vacuum, while in the LQC pre-inflationary dynamics is the non-Bunch-Davies vacuum [104] .

LQC. Perturbations dynamics on quantum geometry In loop quantum cosmology, the phase space of gauge invariant perturbations is spanned by 3 canonically conjugate pairs representing: one the Mukhanov-Sasaki and two the tensor modes,. In co-moving momentum space they are represented by

. (2.12)

Therefore, on using the Mukhanov-Sasaki variables and the two tensor modes denoted by, the quantum fields and propagate on quantum FLRW geometries which are all regular, free of singularity. A key difference with respect to the standard inflation where the fields propagate on a classical Friedmann solution,. Thus, by construction, the framework encompasses the Planck regime and resolves the transPlanckian problem. The evolution equation for tensor modes is given by [104]

(2.13)

where is the momentum conjugate to, , are c-numbers derived from the 4 th order adiabatic regularization that depends only on and are respectively the dressed scale factor and conformal time.

Inflation and slow roll inflation in LQC In LQC [99] [100] , where quantum geometry effects modify the Einstein equations, the Hamiltonian constraint (Equation (2.9)) must be satisfied at any instant of time and the equations of motions given in terms of the variables, provide the evolution

(2.14)

(2.15)

At scales far away from the Planck scale, the quantum effects vanish, the Hamiltonian constraint reduces to the classical Friedmann equation

, (2.16)

and the slow-roll parameters are given by two distinct sets of parameters

. (2.17)

In the context of inflationary models, the WMAP7 data [70] —parameterized assuming that the onset of inflation is in the BD vacuum—are tailored to the co-moving mode (or wave number) given by

(2.18)

where is the today scalar factor and the mode that has just reentered the Hubble radius today. Within each inflationary model the WMAP data (amplitude of the scalar power spectrum, spectral index) constrains the initials values of field describing the homogeneous isotropic background at time at which the co-moving mode exits the Hubble radius during the slow-roll stage and the LQC modifications to general relativity are highly suppressed.

At this time the amplitude of the scalar power spectrum the spectral index the Hubble parameter H and radius are

(2.19)

Further, considering the model with quadratic potential, the equation of motion of the scalar field (Equation (2.13)) simplifies to

. (2.20) s

For this potential, the value of the slow-roll parameter at the time is given by, and at the inflaton, its time derivative, the mass m, and the slow-roll parameter have the following values [100]

(2.21)

This data leads to the slow-roll inflation with

50 - 60 e-folding starting at.

In LQC pre-inflationary dynamics [104] , at the bounce time the value of the background inflaton is constrained in the interval and a solution to the dynamical equations which enters the slow-roll compatible with the seven years WMAP data [70] in its evolution, is obtained for each value. The numerical solutions show that for, none of the modes that are in the observational range will encounter a significant curvature during their pre-inflationary evolution, and then there are no significant excitations over the BD vacuum at the onset of inflation. On the other hand, if, the quantum state at the onset of the slow-roll inflation would carry excitations over the BD vacuum (non-Bunch-Davies vacuum) in modes that are observable in the CMB, where the k_{0} mode in the range that were in the non-BD vacuum leaved their fingerprints—the bispectrum NG flattened configuration.

The LQC scalar power-spectrum is related to the standard inflationary power-spectrum with as the state at onset of inflation

, (2.22)

where the Bogoliubov coefficients and are functions only of and represents the number density of the BD excitations with momentum per unit co-moving volume contained in the state. For modes with lower k values, the LQC power-spectrum has a highly oscillatory behavior and differs from the standard power spectrum while for reference mode used in the WMAP data, the LQC and the standard inflationary predictions are almost the same.

In the standard slow-roll inflation scenario, a significant result is the relation between the tensor-to-scalar ratio and the tensor spectral index . In LQC, the ratio does not depend on the pre-inflationary dynamics, and then while the tensor spectral index is modified in the Planck regime. In [104] , numerical simulations have shown that the imprint left by the pre-inflationary dynamics is potentially observable: a deviation from the standard inflation prediction at low k values.

In summary, LQC and inflationary paradigm predictions are the same in the low energy regime, but drastically differ at Planck scales. Therefore, only the comparison of their key features with the cosmological observations can distinguish between the two, i.e., the excitation over the Bunch-Davies state, non-Gaussianities of the halo bias and the type distortions in CMB.

3. Quantum Fluctuations and Density Perturbations

3.1. Inflation and Cosmological Perturbations

In the inflationary scenario the universe is described by the FLRW solution to Einstein’s equations with a scalar field as matter source, together with small inhomogeneities that are approximated by a first order perturbation. Fourier modes of quantum fields with co-moving wave number in the range are supposed to be initially in the Bunch-Davies vacuum and its quantum fluctuation, soon after any mode exits the Hubble radius, are assumed to behave as a classical perturbation which evolve according to linearized Einstein’ equations. The existence of inflationary perturbation implies that there must be tiny inhomogeneities at the surface of last scattering, and that they serve as seeds which grow into a large scale structure.

In the simplest inflationary model [31] [41] , the average amplitude of the fluctuations (of all wavelengths) of the field generated by the inflation during a typical time interval is given by [41]

. (3.1)

The amplitude of these fluctuations, since the Hubble parameter changes very slow in the inflationary stage, remains almost unchanged for a very long time. Therefore, since the wavelength of fluctuations depends exponentially on the inflation time, the spectrum of inhomogeneities is almost independent of on a logarithmic scale and then is a flat spectrum [41] —HarrisonZeldovich spectrum.

In general, since the field fluctuations lead to a local time delay, of the end of inflation, a local delay of expansion leads to a density increase in a region of the space, , while the density decreases in the regions where the expansion lasts more time. The field fluctuations then, lead to density perturbations that via gravitational instability become later large-scale structures. The amplitude of these perturbations is defined as [5] [12] [41]

. (3.2)

A definition whose derivation is oversimplified since in the inflation stage H and should be calculated at different time for perturbations with different momentum k. But, in the simplest inflation model with potential, where the field changes very slowly during inflation, the quantity remains almost constant over exponentially large range of wavelengths, and then the spectrum of density perturbations is flat and its amplitude is given by [41]

. (3.3)

In the simplest inflation model, the perturbations on scale of horizon were produced at [28] . Hence, using COBE normalization the value of m is, in Planck units. An exact value of m depends on, which in its turn depends slightly on the thermal history of the universe. In short, in the single field slow-roll paradigm, the quantum fluctuations in the inflaton field itself translate, via the Einstein’s equation, into the curvature and density perturbations which grow over time via gravitational instability.

3.2. Quantum Fluctuations and Density Perturbations

In the cosmic inflation, large scale inhomogeneities are related to the restructuring of the vacuum state due to the exponential expansion of the universe. Inflation converts short-wavelength quantum fluctuation of the scalar field into long-wavelength fluctuations, and when the wavelength (k momentum) of a fluctuation exceeds the horizon size, its amplitude is “frozen in” and its nonzero value at the horizon crossing is fixed by the damping due to the friction term (Equation (2.6)). The amplitude of the fluctuation on super-horizon scales remains unchanged for a very long time, whereas its wavelength keeps growing exponentially [3] . The appearance of such frozen fluctuations is equivalent to the appearance of a classical field that does not vanish after having averaged over some macroscopic interval of time. Finally, when the wavelength of these fluctuations reenters the horizon, at some radiation or matter dominated epoch, the curvature (gravitational potential) perturbations, R, of the space-time give rise to matter (and temperature) perturbations via the Poisson equation.

The curvature perturbations are a combination of gravitational potential and energy density perturbations [59]

(3.4)

where, using the perturbed Einstein equation

, (3.5)

the first order (linear) and the second order (non-linear) contribution to the gravitational potential are given by

. (3.6)

Adiabatic perturbations produce a net perturbation in the total energy density and, through the Einstein equations, in the intrinsic spatial curvature. However, as noted in [59] , energy density and curvature perturbations are not gauge-invariant, hence a gauge-invariant variable, , can be used to define such perturbations. For purely adiabatic perturbations the curvature perturbation is conserved on large scales, and then is appropriate to characterize the amplitude of adiabatic perturbations. At second order the curvature perturbation on large scales evolves due to non-adiabatic pressure perturbations.

The uniform energy density curvature perturbation is related to the comoving curvature perturbation by

, (3.7)

and then on large scales.

In the single-field slow-roll models of inflation, where the intrinsic entropy perturbation of the inflaton field is negligible on large scales, the power-spectrum of the curvature perturbation on large scales is given by [59]

(3.8)

where the asterisk stands for the epoch in which a perturbation mode leaves the Hubble horizon during inflation and the spectral-index of the curvature perturbation to lowest order in the slow-roll parameters, , , is

. (3.9)

The power spectrum of gravity-wave mode is given by

, (3.10)

where is the tensor spectral-index. Therefore, since the fractional change of the power-spectra with scale is much smaller than unity, the power-spectra can be considered as constant on scales relevant for the CMB anisotropy and the tensor—to-scalar amplitude ratio can be defined as

. (3.11)

For or, given in [105] , the upper bound on the energy scale of inflation is given by [59]

(3.12)

In summary, in the single field slow-roll scenario, the density perturbations due to the quantum fluctuations in the inflaton field itself translate, via the Einstein’s equations, into the curvature and density perturbations which grow over time via gravitational instability. High density regions continuously attract more matter from the surrounding space, the high density regions become more and more dense in time while depleting the low density regions. At late times the highest density regions peaks collapse into the large structure of the universe, whose gravitational instability effects are observed in the clustering features of galaxies in the sky.

Inflationary dynamics then, explain the formation of all structure in our universe: from primordial quantum fluctuations to large-scale structure. In this context, since large classes of models indicate the inflationary mechanism as responsible of generation of the curvature fluctuations (Gaussian and adiabatic) on super horizon scales, the power spectrum of the curvature perturbations, its spectral-index, , and the tensor—to-scalar amplitude ratio, r, allow to compare their predictions and key parameters with the current cosmological observations—the Wilkinson Microwave Anisotropy Probe (WMAP) and Planck datasets.

3.3. Fluctuation Dynamics and Power Spectra

The quantum fluctuations in the inflaton and in the transverse and traceless part of the metric, which are amplified by the exponential expansion, yield the scalar and tensor primordial power spectra. In the simple formalism, given in [106] -[108] and used in [74] , the gauge-invariant variable Q describes the inflaton fluctuation in the uniform curvature gauge. The evolution of its mode function is governed by the equation

, (3.13)

where and k is the comoving wave number of the Fourier mode under consideration. The gaugeinvariant field fluctuation is related to the co-moving curvature perturbation, which at large scales is related to the uniform energy density curvature perturbation ζ:.

In a like approach, the gravitational waves are described by two polarization states of the transverse traceless part of the metric fluctuations amplified by the exponential expansion. The evolution equation for their mode function is given by

. (3.14)

The above linearized equations, since the primordial perturbations are small, of order 10 , provide a satisfactory description for the generation and evolution of perturbations in the inflation stage, from the sub-Hubble to super-Hubble evolution.

The power spectra of curvature and tensor perturbations on super-Hubble scales, yielded by the primordial quantum fluctuations in the inflaton and in the transverse traceless part of the metric, are defined as [74]

(3.15)

where, are the scalar and tensor amplitude, , are the scalar and tensor scalar index, the derivatives, are the running of the scalar and tensor spectral index and the asterisk stands for the value of the inflaton field at which the mode crosses the Hubble radius for the first time.

In the single field slow-roll approximation, the number of e-folds before the end of inflation, at which the pivot scale exits the from the Hubble radius, is

, (3.16)

where prime denotes derivatives with respect to the field, and subscript e denotes the end of inflation. The scalar and tensor amplitude, and spectral index, , are given by [74]

(3.17)

where the tensor to scale modes ratio r and the slow-roll parameters, , , are

(3.18)

3.4. Alternative, Equivalent Definitions

In [69] , the quantity closely related to the amplitude, of the primordial curvature perturbations, is considered as a powerful and practical tool to test significant features of a large class of models which predict that is nearly a power law

(3.19)

where gives an approximate contribute of (at a given scale per logarithmic interval in k) to the total variance of, as, while the spectral index and the spectral index are used to define the tilt of the spectrum and the running index. For and the power-law spectrum is a pure Harrison-Zeldovich “scale invariant spectrum”, while for this spectrum is strongly disfavored.

The primordial gravitational waves, predicted by many cosmological models, leaves their signatures imprinted on the CMB temperature anisotropy (the tensor E-mode, B-mode and TE-mode polarization) [69] . The amplitude and power spectrum of these gravitational waves, since gravity is encoded in the metric, are defined as

(3.20)

where is the power spectrum of tensor metric perturbations, h_{k}, the amplitude is given as and the amplitude, in the form of the tensor to scalar ratio, r, is defined as [69] [70]

(3.21)

which is used to calculate the tensor mode contributions to the B-mode, E-mode and TE power spectra. The spectral index and the ratio r are related to the inflation slow-roll parameters, and then can be used to constraint the inflationary models [23] [69]

, (3.22)

(3.23)

where is the reduced Planck mass, and the derivatives are evaluated at the mean value of the scalar field at the time that a given scale leaves the Hubble horizon. The combination of these two equations relates and r

. (3.24)

This equation, since r is correlated to both and the curvature of the potential allows to distinguish models on the plane and justify the use of the sign and magnitude of the potential curvature to classify inflation models [69] .

In the period of cosmic inflation, the generated quantum fluctuations, as explained in (3.2), became the seeds for the cosmic structures observed today. In this contest, inflation models predict that the primordial fluctuations is nearly a Gaussian distribution with random phase. This prediction is based on the assumption that the probability distribution of quantum fluctuations, of the free scalar field in the Bunch-Davies vacuum is a Gaussian distribution, so that also the probability distribution of primordial curvature perturbations, R, generated by the field would be a Gaussian distribution. Therefore, deviations from a Gaussian distribution, as nonGaussian correlations in the primordial curvature perturbations, is a significant test of the inflation predictions.

Today, the WMAP and Planck results, and the non-Gaussianity (NG) of the primordial fluctuations, captured by the bispectrum different configurations (local, equilateral, folded, orthogonal), are the most important information used to rule out or constrain the 193 inflation models available [109] [110] .

The primordial curvature perturbation, R, cannot be measured directly, a more observable quantity is the curvature perturbation during the matter era, written in the local form

, (3.25)

where which represents the gravitational potential at linear order, is correlated to the linear part of the curvature perturbation, is the “local nonlinear coupling parameter”, and both sides of the equation are evaluated at the same location in space [69] [70] .

The amplitude of non-Gaussian correlations can be evaluated using the angular bispectrum of CMB temperature anisotropy, the harmonic transform of the 3-point correlation function. The observed angular bispectrum is related to the 3-dimensional bispectrum of primordial curvature [73]

, (3.26)

where the potential equivalent to the Bardeen’s gauge-invariant gravitational potential is defined in terms of the co-moving curvature perturbation ζ on super-horizon scales by. The bispectrum measures the correlation among three perturbation mode and, for translational and rotational invariance, it depends only on the magnitude of the three wave-vectors

, (3.27)

where the “nonlinearity” parameter, , is a dimensionless parameter which measures the amplitude of NG.

3.6. Non Gaussian Configuration

Different NG configurations are linked to different mechanisms for the generation of non-Gaussian perturbations at different scales, where the primeval inflaton field interactions, encoded in the three wave numbers, , are captured by signal peaking in triangles of different form: Orthogonal, Local, Equilateral, Folded, [73] .

Orthogonal NG, which generate a signal with a positive peak at the equilateral configuration and a negative peak at the folded configuration [69] .

Local NG, where the signals peaks in squeezed triangles, which occurs, not only when the primordial NG is generated on super-horizon scales, as in the multi-field inflation models [111] -[113] and in a new bouncing cosmology [114] , but also when enhanced NG is generated by the modified non-Bunch-Davies (NBD) vacuum initial state, as in the excited canonical single-field inflation [115] .

Equilateral NG triangular configuration of the three comoving momentum vectors due to correlation between fluctuation modes of comparable wavelength, which occurs if the three perturbation modes cross the horizon at approximately the same time.

Examples of this class include: single field with non-canonical kinetic term [60] [61] , k-inflation [62] , DiracBorn-Infeld (DBI) inflation [63] [64] , models with higher-derivative interactions of the inflaton field as ghost inflation [58] , effective field theory models [116] .

Folded NG, where the signal peaks in flattened triangles, which occurs when the primordial NG is generated by the initial state, where the inflaton field interactions are stronger at the onset of inflation than at the end, and then there is enhanced NG coming from excited initial state in the early inflation stages.

Examples of this class include: single field models with non-Bunch-Davies (NBD) vacuum [61] [65] , models with general higher-derivative interactions [68] , excited canonical single-field inflation model with modified NBD initial state whose enhanced bispectrum has two leading order shapes [115] : the single-field NBD1 squeezed configuration and the single-field NBD2 flattened configuration which differ from the much flattened non-Bunch-Davies (NBD) bispectrum [61] . These shapes are typically oscillatory since are regularized by a cutoff scale giving the oscillatory period. This cutoff, where is the speed of the sound of inflaton fluctuations, is determined by the finite time in the past when the NBD component was initially excited.

Anyway, all these bispectrum differ from that for single-field inflation models with a Bunch-Davies vacuum initial state, particularly the NBD1 squeezed shape whose wave number is of importance for observations [117] .

[The BD vacuum is a state tailored to a few e-folding before the mode leaves the Hubble horizon in this state the physical length of the mode equals the radius of the observable universe at the surface of last scattering].

Thus, the bispectrum defined in Equation (3.27) measures 1) the fundamental self-interactions of the scalar field (or fields) involved in the inflation, as well as 2) their interactions which generate the primordial curvature perturbations, and also 3) the non-Gaussianity—the quantities —which represent the higher order perturbations occurring during or after the inflation. Therefore, it brings insights into the physics behind inflation, as the pre-inflationary dynamics of the loop quantum cosmology [104] .

4. Cosmological Models and Current Observations

4.1. The WMAP and Planck Probes Observations

Today, the Wilkinson microwave anisotropies probe (WMAP) data [69] -[71] , the Planck measurements of the cosmic microwave background (CMB) [72] -[74] , and their search of non-Gaussianity in the distribution of dark matter and galaxies in the late universe allow to compare the key parameters and predictions of many cosmological models with the observational data.

In the inflationary scenario the CMB, whose amplitude of tensor B-mode polarization is proportional to the energy of inflation and tied to the range of inflaton field, is the unique window between the late-time universe and the early universe at the time of the potential energy domination that drove cosmic inflation. The phenomenology of non-Gaussianity in the distribution of dark matter and galaxies allows to study the dynamics of inflation. It is then clear that a wide range of cosmological data plays an essential role 1) in testing the large classes of model and their predictions and 2) in constraining their key parameters.

4.2. The WMAP Datasets: ΛCDM Model

In the standard cosmological model (ΛCDM), based on a flat expanding universe whose constituents are dominated by cold dark matter (CDM) and a cosmological constant (Λ) at late times, the dynamics is governed by general relativity and the primordial seeds of structure formation are nearly scale-invariant adiabatic Gaussian fluctuations.

The six key parameters of the simplest ΛCDM model have proven to be a satisfactory fit to nine years WMAP data alone, or combined with the latest distance measurements from baryon acoustic oscillation (BAO) in the distribution of galaxies, the Hubble parameter H_{0} measurements, and the extended CMB data (eCMB) which include the Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT) data. Here we report only the nine years WMAP combined (WMAP + eCMB + BAO + H_{0}) data given in ([71] , Table 2 ).

The given WMAP results show that the six parameters ΛCDM model describes all the data remarkable well, and there are no evidence for deviations from this model: the geometry of the observable universe is flat and the dark energy is consistent with a cosmological constant. The amplitude of matter fluctuations, , is consistent with all existing data, including cluster abundance, peculiar velocity, and gravitational lensing. The simplest ΛCDM model then, survived the most stringent nine years WMAP test.

Beyond the ΛCDM parameters, the combined (WMAP + eCMB + BAO + H_{0}) data exclude a scale invariant primordial power spectrum at significance. In addition to adiabatic fluctuations, where all species fluctuate in phase and then produce curvature fluctuations, isocurvature perturbations producing no net curvature are possible. These entropy, or isocurvature perturbations—occurring when an over-density in one specie compensate the under-density in another—have a measurable effect on the CMB shifting the acoustic peaks in the power spectrum. But, no significant contribution from these perturbations has been detected in the nine year WMAP data, whether or not additional data have been included in the fit.

4.3. The WMAP Datasets: Inflation Models

In the combined (WMAP + eCMB + BAO + H_{0}) data for a power-law spectrum of primordial curvature perturbations

, (4.1)

With the power-law spectral index, without B-mode polarization measurements—that is ignoring tensor mode, is

(4.2)

The tightest constraint on tensor modes r (Equation (4.3)) and on the running scalar index from the nine year WMAP combined data are [71]

. (4.3)

Therefore, the current data strongly disfavor a pure Harrison-Zeldovich spectrum, also if tensor modes are included in the model fits.

In the large classes of inflation models, the WMAP data [71] well support the predictions of single-field inflation models for features of primordial curvature: the temperature fluctuations, which linearly trace the primordial curvature perturbations are adiabatic and Gaussian [118] . The limits on primordial gravitational waves are consistent with many inflation models, including the Starobinsky model [8] with spectral index and tensor to scalar ratio r given by

(4.4)

where N is the number of e-folding related to and r. The non-Gaussian features of primordial fluctuations in inflationary models [29] [30] has not been evidenced. In general, since an analysis of the CMB maps [118] has shown that all the quantities does not differ significantly from zero, there is no compelling evidence for deviations from Gaussianity. Hence, inflationary models predicting the existence of no-Gaussianity are strongly disfavored by the WMAP results.

The two dimensional joint marginalized constraints (68%, 95%) on the primordial tilt allows to constrain and distinguish inflation models on plane [70] . In this class of models, as reported in [69] , the single-field inflation models with chaotic potential with, have different fate. The model with potential is far away from the 95% region for both N = 50 and 60, and then is excluded at more than 99% CL. The inflation model with potential lies outside of the 68% region for N = 50, while for N = 60 is at the boundary of this region. Hence the model is under pressure.

In conclusion, the six parameters ΛCDM model and the single-field inflation models for Gaussian features of primordial curvature survived the most stringent WMAP tests.

In the Plank 2013 results [72] the most important conclusion is that the Planck temperature power spectrum at high multipoles agrees with the predictions of the six-parameter ΛCDM model. Here we report only the (Plank + WP 68% limits) data given in ([72] , Table 2 , Table 11 for).

The Planck data show that the simplest ΛCDM model provides a good match to the Planck power spectrum of the lensing potential, and to the TE and EE power spectra at high multipoles, but no in the low multipoles range This mismatch, and other anomalies at low multipoles, although not of decisive significance for the model, imply that it may be incomplete.

Moreover, in the Planck 2013 data given in ([72] , Table 11 ) the low value of the nine years WMAP Hubble parameter, H_{0} = 74 ± 11, with respect the Planck + WP H_{0} = 65.2 ± 1.8, and a high matter density, Ω_{m} = 0.315 ± 0.016 are in tension. Therefore, as noted in [119] , if the tension between previous measurement of H_{0} and that recently derived by the Planck team within the assumptions of a six-parameter flat ΛCDM model (including tension with WMAP 9), cannot be resolved by unknown systematics, it will force the rejection of the six-parameter model in favor of a more complex alternative.

However, despite these limits, the six parameters ΛCDM model has proven to be successful in describing a wide range of cosmological data, including the Type Ia supernovae magnitude-distance relation, baryon acoustic oscillation measurements, the large-scale clustering of galaxies and cosmic shear, and its predictions have been confirmed but the nine years WMAP and Planck 2013 results. Hence, the tension between the WMAP and Planck data are not a compelling evidence of its incompleteness. Other observations are needed to decide about its limits.

4.5. The Planck Datasets: Non-Gaussianity

In the Planck results 2013 [73] , dedicated to constraints on primordial non-Gaussianity, the final results

and

(4.5)

show that there is no evidence for primordial NG in these shapes. Therefore, the models predicting the non Gaussianity of the primeval perturbations via these bispectrum shapes are strongly disfavored or constrained.

Here, we give a partial list of models. 1) For the non Bunch-Davies models with flattened bispectral often in combination with non trivial squeezed limit, which can be generated by strong disturbance away from background slow-roll evolution or additional trans-Planckian physics [61] [65] [115] [120] , there is no convincing evidence for the predicted bispectra and the related non-Gaussianity of the primordial perturbations. The same results hold for the non-Bunch-Davies resonant models.

2) Warm Inflation [52] [53] produce a related shape with a sign change in the squeezed limit. The resulting bispectrum has no evidence for significant correlation with Planck data.

3) In the ekpyrotic/cyclic scenarios [49] -[51] the “ekpyrotic conversion” mechanism has been ruled out and the parameter space of the “kinetic conversion” mechanism significantly limited.

However, as noted in [73] , the results for the non-Bunch-Davies (NBD) models are just preliminary. A systematic search for best-fit Planck NBD models using the parameter freedom available has to be undertaken.

Anyway, a possible explanation for the non-detection of the non-Gaussianity in the NBD models is given in [121] for open inflation. The amplitude of the bispectrum is exponentially suppressed in the sub-curvature region for wavelengths shorter than the today curvature radius, and so the non Bunch-Davies effect of open inflation on the observable bispectrum is exponentially suppressed.

4.6. The Planck Datasets: Cosmic Inflation

In the Planck 2013 results [74] , dedicated to the status of cosmic inflation, the scalar amplitude A_{s} and the scalar spectral index n_{s} (adding a tensor component) for the primordial curvature perturbations spectrum (Equation (3.15)) are

, (4.6)

a departure from The tightest constraint on tensor modes r and on the running scalar index are

. (4.7)

Here, we just mention some of the more than hundred inflationary models [110] .

1) In the Planck 2013 results for cosmic inflation [74] , the simplest slow-roll single field inflation survived an exacting test with the Planck data. The spatial curvature at 95% CL and the bispectral non-Gaussianity parameter consistent with zero are compatible with the zero spatial curvature and small value of as predicted in this simplest slow-roll single field model.

2) The axion monodromy potentials, and models [109] [122] are the most compatible with the Planck data, while the model is decisively ruled out, the large quadratic potential and natural inflation, are somewhat disfavored by the Planck data, when broader entropy generation scenarios are considered.

3) Other models with an exponential potential, a monomial potential with a power larger than two, or hybrid model driven by a quadratic term are disfavored at more than 95% CL. The axion and curvaton scenario in which the isocurvature mode is uncorrelated or fully correlated with the adiabatic mode, are disfavored by Planck data. The IR DBI (Dirac-Born-Infeld) inflation, k-inflation, inflation involving gauge fields and warm inflation have been strongly constrained. In the ekpyrotic/cyclic scenarios the “ekpyrotic conversion” mechanism has been ruled out and the parameter space of the “kinetic conversion” mechanism strongly limited.

In summary, in the simplest versions of the chaotic inflation with for the predictions of the inflationary models with are consistent with the Planck data, the simplest quadratic potential is marginally consistent, whereas the model is ruled out. In the chaotic inflation in super-gravity [123] the predictions of the model with potential

(4.8)

are in agreement with the Planck data. Therefore, as noted in [123] , a slight modification of the simplest chaotic inflation with a quadratic potential leads to a model consistent with the Planck 2013 results.

A Long list of other models compatible with the Planck data is given in [124] where, besides the Bayesian analysis of the inflationary models consistency with the Planck results, is noted that the models consistent with the Planck results cannot be considered as “true” models, they are the simplest and most effective inflationary hypothesis compatible with the Planck 2013 CMB data.

Before to conclude this section, we recall the predictions of the loop quantum cosmology and discuss their consistency with the Planck 2013 data.

In the LQC application, for the simplest slow-roll inflation with potential [100] [104] , the LQC and the simplest inflation model predictions are the same in the low energy regime, but strongly differ at the Plank scale. In the simplest inflationary paradigm, the state at the onset of inflation state is the Bunch-Davies vacuum. In the LQC pre-inflationary for at the bounce the quantum state at the onset of the slowroll inflation is the non-bunch-Davies vacuum [104] .

Therefore, since in the Planck 2013 results on the non-Gaussianity [73] there is no evidence for primordial non-Gaussianity (Equation (4.4)), the LQC pre-inflationary dynamics predictions are not consistent with the Planck data, but for its key features and predictions are compatible with the observations.

Hence, this LQC specific application for the inflation with is under pressure for its pre-inflationary dynamics predictions, but the novel features of the loop quantum cosmology—space and time discrete at the Planck scale, the modified Einstein equation (Equation (2.9)) which replace the singularity with a quantum bounce, and others—are still significant and innovative.

However, as noted in [73] , the results for the non-Bunch-Davies (NBD) models are just preliminary. A systematic search for best-fit Planck NBD models using the parameter freedom available has to be undertaken.

In conclusion, the Planck 2103 data for non Gaussianity and for inflationary models (Equations (4.6), (7)), including the Bayesian consistency, are not definitive since future observations can change the constraint limits. Therefore, models that are disfavored or severely constrained by the current Planck data can be favored, or more disfavored, by future observation, i.e., in ([73] , p. 10): “The model with a quadratic potential, n = 2 (Linde, 1983), often considered the simplest example for inflation, now lies outside the joint 95% CL for N 60 e-folds”.

5. Cosmological Models: Unsolved Fundamental Questions

5.1. The Status of our Universe

Today, the six-parameters ΛCDM model and the simplest slow-roll single field inflation, which are the most effective hypothesis compatible with the WMAP and Planck 2013 data, provide a convincing description of the status of our universe and of the large-scale formation via primordial quantum fluctuations and gravitational instability. The ΛCDM model describes successfully many features of the evolution of the universe—the Hubble expansion, the existence of a CMB with a blackbody spectrum, the primordial D, 3 He and 7 Li abundance, the sum of the masses of the three families of neutrinos, the dark energy equation of state parameter, the number of effective relativistic species, the big bang nucleosynthesis.

In the single field slow-roll paradigm, the density perturbations due to the quantum fluctuations in the inflaton field itself translate, via the Einstein’s equations, into the curvature and density perturbations which grow over time via gravitational instability. High density regions continuously attract more matter from the surrounding space, the high density regions become more and more dense in time while depleting the low density regions. At late times the highest density regions peaks collapse into the large structure of the universe, whose gravitational instability effects are observed in the clustering features of galaxies in the sky. Thus, the origin of all structure in the universe probably comes from an early era where the universe was filled with a heuristic scalar field generated by the vacuum fluctuations during inflation, and nothing else.

5.2. The Unsolved Questions

The inflationary paradigm has successfully explained the generation of primordial perturbation and of gravitational waves. However, despite its successes, it is conceptually incomplete in several respects. The inflationary space-time is past incomplete, and then inherit the big bang singularity. The initial conditions are specified at the onset of inflation and then evolves the quantum perturbations, but at the onset of inflation the curvature is some 10 - 10 times the Planck scale. Thus, what really happened at the Planck regime is bypassed. Another question is the quantum to classical transition [125] -[127] . The inflationary physics replaces the quantum state of perturbations with a Gaussian statistical distribution of classical perturbations, but what happens physically is not explained.

Furthermore, the inflationary paradigm, as well as the ΛCDM model and also the loop quantum cosmology, does not tell where the inflaton with mass-energy above 10 12 Gev comes on, nor how the potential arise. The hypothesis of its generation from vacuum fluctuation does not explain what physically happens in the early era. A crucial question then the origin of the inflaton, a question closely related to the problem of the origin of our universe, but its solution is beyond the existing theories. In the same Standard Model of particles physics, whose 26 free parameters justifies that is only a low energy effective theory, an approximation, an outstanding question is “why our universe is matter dominated”, “what is its origin?” [128] .

In our days, the available understanding of the physical world is fragmented, leaves unanswered fundamental questions, and then the need of a new synthesis is the great challenge of fundamental physics. It is not then a case, if on the crucial question of the universe origin the relativistic and quantum cosmological theories are still confined in the old paradigms—the eternal, endless sequence of epochs and the quantum creation from nothing. Today, the deepest mysteries of our universe is yet the puzzle of whence it came [129] .

In our days, we live in an unusual time, perhaps a golden age of the cosmology. The impressive cosmological observations and measurements have revealed, not only that the observable part of the universe is nearly flat, filled with photon, baryons (4%), dark matter (23%) and dark energy (73%), but also that the expansion of the universe is accelerating: a pair of galaxies separated by a million light years are drifting apart at a velocity of 68 kilometers a second as the universe expands. An universe then, that is expanding more faster than in the past and consists primarily of mysterious exotic substances, defined generically as dark matter and dark energy since their real nature is yet unknown. These observations have deeply changed the key features of the universe and put under pressure many of the more than hundred cosmological models available.

The WMAP and Planck 2013 results have already ruled out many models and significantly constrained others. Their severe screening of large classes of models has ruled out the inflationary model and others, marginalized the simplest quadratic potential model, disfavored the ekpyrotic/cyclic model, already under strong pressure after the discovery of the universe acceleration and the detection of primordial gravitational waves, and favored both the six-parameters ΛCDM model and the simplest slow-roll single field inflation, which have survived their stringent tests.

Thus, in the landscape of more than hundred model, the standard cosmological model combined with the single field inflationary model is one the most effective hypothesis which provides a convincing description of the status of the observable universe and of the large-scale formation via primordial quantum fluctuations and gravitational instability.

The cosmological observations then, have significantly restricted the number of models compatible with their datasets and, beyond their tests on different classes of models, have deeply affected the development of new cosmological theories. Their main purpose is no more the construction of hypothesis and models to describe and predict large local non-Gaussianity, but the elaboration of new models of inflation capable to fit their predicted data in the plane provided by the current and future cosmological observations, as WMAP, Planck and others.

In conclusion, the standard cosmological model and the simplest single field slow-roll inflation well-fit the current WMAP and Planck 2013 datasets. Thus, if this combination will survive future observations, is likely to be the one chosen by Nature: the more than 350 millions cosmic structure, catalogued in the SDSS (Sloan digital sky survey) and others archives, come from an early era where the universe was filled with a scalar inflaton field and nothing else. Future observations will decide the fate of this heuristic conclusive hypothesis.

## Problem poser

If the expansion of the universe is accelerating due to dark energy, then by the time the photons climb out, the supercluster has expanded, and its gravity is a little less strong. So the photons exit relatively easily and with more energy than they had when they entered the gravitational well.

But photons going through a void actually lose energy, ending up colder than if they had been flying through a series of superclusters. Rudnick thinks that the discovery of the void ties in neatly with the WMAP cold spot and the existence of dark energy. “What the community says remains to be seen,” he told **New Scientist**. “People will take shots at it now.”

Because the CMB is leftover radiation from the big bang, some cosmologists have said that the cold spot is a problem for the theories of the early universe. But Rudnick says that the void could have been created billions of years after the big bang. “We have taken the problem away from the very early universe and put the problem in the time of structure formation,” he says.

Computer simulations that recreate the formation of clusters and super-clusters have never seen voids of this size. That could be because modellers have not simulated large enough volumes to see such a void, says Rudnick. If they did, maybe a void would emerge. “It is an open question whether this will create problems for structure formation,” he says.

## Why Isn't Our Universe Perfectly Smooth?

The stars and galaxies we see today didn't always exist, and the farther back we go, the closer to . [+] perfectly smooth the Universe gets, but there is a limit to the smoothness it could've achieved, otherwise we wouldn't have any structure at all today. To explain it all, we need a modification to the Big Bang: cosmological inflation.

NASA, ESA, and A. Feild (STScI)

When we examine our Universe, looking out at the planets, stars, galaxies, and vast cosmic voids separating them, "smooth" isn't exactly the first word that comes to mind. The enormous cosmic web is one of the clumpiest things imaginable in the Universe, with a planet like Earth some 10 30 times denser than average. Yet the Universe wasn't always this clumpy, or it wouldn't have evolved to appear the way we see it today. It had to have been born almost perfectly smooth, where the imperfections were just a few parts in 100,000, or it wouldn't have taken hundreds of millions of years to form the first galaxies. Yet those tiny imperfections were vital, or we wouldn't have formed the structure we see today at all! After centuries of not understanding how this happened, one of cosmology's most controversial theories, inflation, provided the answer. And now that our measurements have achieved unprecedented precision, its predictions check out spectacularly.

A visual history of the expanding Universe includes the hot, dense state known as the Big Bang and . [+] the growth and formation of structure subsequently. However, in order to get the structure we see today, the Universe couldn't have been born perfectly smooth.

According to cosmic inflation, the hot Big Bang wasn't the very beginning of space and time, but was merely a hot, dense, rapidly expanding early state. It was cosmic inflation, a phase where the Universe was dominated not by matter and radiation, but by the energy inherent to space itself, that set up the Big Bang. This inflationary phase was characterized by an exponential expansion of space, where the Universe doubled, then quadrupled, then octupled (etc.) in size as time went on. After as little as 10 -33 seconds, a region the size of a theoretical string from string theory would have been stretched to a scale larger than the observable Universe is today. In other words, cosmic inflation takes whatever existed beforehand and stretched it really, truly, and perfectly flat-and-smooth.

Inflation causes space to expand exponentially, which can very quickly result in any pre-existing . [+] curved or non-smooth space appearing flat. If the Universe has any curvature to it at all, it has a radius of curvature hundreds of times larger than what we can observe.

E. Siegel (L) Ned Wright’s cosmology tutorial (R)

This seems, at first glance, to pose a tremendous problem. If inflation stretches space to be flat, uniform, and smooth, indistinguishably so from perfection, then how did we arrive at a clumpy Universe today? Both Newton's and Einstein's theories of gravity are unstable against imperfections, meaning that if you start with an almost-but-not-quite perfectly smooth Universe, over time, the imperfections will grow and you'll wind up with structure. But if you start with perfect smoothness, with literally no imperfections, you're going to remain smooth forever. Yet this doesn't jibe with the Universe we observe at all it had to have been born with imperfections in its matter density.

A map of the clumping/clustering pattern that galaxies in our Universe exhibit today. The . [+] requirement to get there are initial imperfections in the matter/energy density.

Greg Bacon/STScI/NASA Goddard Space Flight Center

This naive picture of inflation must therefore be incomplete. There has to be some way of generating these imperfections, otherwise the Universe wouldn't exist the way we see it. But an important property of the Universe, and of inflation, comes to the rescue in the most spectacular of ways. You see, empty space itself isn't perfectly flat and smooth on its own, but rather, at the smallest scales, exhibits quantum fluctuations.

Visualization of a quantum field theory calculation showing virtual particles in the quantum vacuum. . [+] Even in empty space, this vacuum energy is non-zero.

This can be viewed in many ways: an inherent uncertainty to the energy of space itself as vacuum fluctuations or as sets of particle-antiparticle pairs popping in-and-out of existence. But regardless of how you view it, one thing remains clear: if you were to graph the energy density of the Universe, and look at it on extremely small and granular scales, you'd see that it wasn't uniform and constant in space or time, even if you removed all the matter and radiation from it. There are quantum fluctuations inherent to the fabric of space itself.

An illustration of the early Universe as consisting of quantum foam, where quantum fluctuations are . [+] large, varied, and important on the smallest of scales.

Normally, these fluctuations cancel each other out, on average, and so you just wind up with a tiny zero-point energy that's positive inherent to space itself. But during inflation, these quantum fluctuations don't have the opportunity to average out, because space itself is expanding at this exponential rate!

Instead, what happens is that these fluctuations get stretched across the Universe, and so the idea of a quantum fluctuation no longer is restricted to a very small scale. In timescales that are only a tiny fraction-of-a-second long, these quantum effects can get stretched to be fluctuations in energy on stellar, galactic, or even Universe-encompassing scales!

The quantum fluctuations that occur during inflation do indeed get stretched across the Universe, . [+] but they also cause fluctuations in the total energy density, leaving us with some non-zero amount of spatial curvature left over in the Universe today. These field fluctuations cause density imperfections in the early Universe, which then lead to the temperature fluctuations we experience in the cosmic microwave background.

E. Siegel / Beyond the Galaxy

As inflation continues, new quantum-scale fluctuations get created, resulting in additional, smaller-scale fluctuations superimposed atop the larger-scale ones. This goes on and on, creating a pattern of fluctuations, and random regions of all sizes that have overdense and underdense energy densities, for as long as inflation goes on.

Then, after an indeterminate amount of time, inflation comes to an end. And when this occurs, all of that energy inherent to space itself gets converted into matter, antimatter, and radiation. As inflation ends, the hot Big Bang begins, and the Universe becomes filled with stuff.

The analogy of a ball sliding over a high surface is when inflation persists, while the structure . [+] crumbling and releasing energy represents the conversion of energy into particles.

But in the regions that were initially overdense to begin with in terms of energy, due to those quantum fluctuations during inflation, a tiny bit more matter, antimatter, and radiation than average will come to exist in those places. In regions that were underdense, a bit less-than-average matter, antimatter, and radiation will come to exist there. And this spectrum over overdensities and underdensities should result in ever-so-slightly cooler and hotter regions, in terms of temperature, in the Universe as a result.

Regions of space that are slightly denser than average will create larger gravitational potential . [+] wells to climb out of, meaning the light arising from those regions appears colder by time it arrives at our eyes. Vice versa, underdense regions will look like hot spots, while regions with perfectly average density will have perfectly average temperatures.

E. Siegel / Beyond The Galaxy

After the Universe has been around for a little while, expanding and cooling, gravitation gets to work. This grows the fluctuations that existed in whatever direction they departed from the average. The slightly hotter regions, being underdense, will more easily give up their matter to denser regions. The colder regions, being overdense, will preferentially attract matter more efficiently than underdense or average-density regions will.

There's an intricate balance between gravitation, which works to attract everything according to the logic above, and radiation, which presses back against regions that become too dense too quickly. It's this interplay of forces, between gravitation, radiation, and the initial fluctuations from inflation, that give rise to the bumps, wiggles, and imperfections that we see in the cosmic microwave background.

The fluctuations in the CMB are based on primordial fluctuations produced by inflation. In . [+] particular, the 'flat part' on large scales (at left) have no explanation without inflation, and yet the magnitude of the fluctuations constrains the maximum energy scales the Universe reached at the end of inflation. It's far lower than the Planck scale.

The initial fluctuations, on average, must have had a mean value of 1-part-in-30,000 or so, which is how we arrive at the fluctuations we observe in the Big Bang's leftover glow. These fluctuations then grow, after the Universe becomes neutral and the radiation stops scattering off of electrons, to produce the large-scale structure we see in the Universe today. Over time, this leads to gravitational growth into stars, galaxies, clusters, and the great cosmic voids separating them.

A detailed look at the Universe reveals that it's made of matter and not antimatter, that dark . [+] matter and dark energy are required, and that we don't know the origin of any of these mysteries. However, the fluctuations in the CMB, the formation and correlations between large-scale structure, and modern observations of gravitational lensing all point towards the same picture, originating from cosmic inflation.

Chris Blake and Sam Moorfield

If the Universe were born perfectly smooth, there would be no way to obtain the detailed structure, on both large scales and small ones, that we have today. Our observations require that, somehow, fluctuations of the same magnitude exist on all scales, and that the Universe needed to be born this way. When inflation was first theorized in the late 1970s and early 1980s, there was no way of knowing how these fluctuations would turn out this was a prediction that inflation made that wouldn't be verified for decades! Yet the confirmation here is spectacular, as no other theory has a way of generating these fluctuations, and the observations matched what inflation predicted in perfect, incontrovertible fashion as satellites like COBE, WMAP, and most recently, Planck, returned their data.

The quantum fluctuations that occur during inflation get stretched across the Universe, and when . [+] inflation ends, they become density fluctuations. This leads, over time, to the large-scale structure in the Universe today, as well as the fluctuations in temperature observed in the CMB.

E. Siegel, with images derived from ESA/Planck and the DoE/NASA/ NSF interagency task force on CMB research

The result is a story so compelling and in agreement with the data that there's practically no alternative. Inflation isn't just the thing that happened to set up the Big Bang or solve a slew of problems that we knew beforehand it made quantitative predictions about what we could expect to exist in the Universe, from early times to modern ones, and observations have confirmed it. Inflation, and its quantum nature, is the reason why the Universe isn't perfectly smooth today, and that's a very good thing. Without it, it never would have been possible for us to exist.

## Subaru Telescope Helps Create the Most-Extensive Map of Neutral Hydrogen Gas in the Early Universe

The distribution of galaxies in the proto-supercluster region 11.5 billion years ago (top left), and the Subaru Telescope Suprime-Cam image used in this work (right, larger image). Neutral hydrogen gas distribution is superposed on the Subaru image. The red color indicates denser regions of the neutral hydrogen gas. Cyan squares correspond to member galaxies in the proto-supercluster, while objects without cyan squares are foreground galaxies and stars. The distribution of neutral hydrogen gas does not align perfectly with the galaxies.

*Using the Subaru Telescope, astronomers create the most-extensive map of neutral hydrogen gas in the early universe.*

Scientists have used the Suprime-Cam on the Subaru Telescope to create the most-extensive map of neutral hydrogen gas in the early universe. This cloud appears widely spread out across 160 million light-years in and around a structure called the proto-supercluster. It is the largest structure in the distant universe, and existed some 11.5 billion years ago. Such a huge gas cloud is extremely valuable for studying large-scale structure formation and the evolution of galaxies from gas in the early universe, and merits further investigation. The team included scientists from Osaka Sangyo University, Tohoku University, Japan Aerospace Exploration Agency (JAXA) and others.

“We are surprised because the dense gas structure is extended much more than expected in the proto-supercluster,” said Dr. Mawatari. “Wider field observations with narrow-band filters are needed to grasp full picture of this largest structure in the young Universe. This is exactly the type of strong research that can be done with Hyper Suprime-Cam (HSC) recently mounted at the Subaru Telescope. We intend to study the gas – galaxy relation in various proto-superclusters using the HSC.”

**Understanding Matter Distribution in the Universe**

Stars assembled to form galaxies, and galaxies are clustered to form larger structures such as clusters or superclusters. Matter in the current universe is structured in a hierarchical manner on scales of

100 million light-years. However, we cannot observe inhomogeneous structure in any direction or distance over scales larger than that. One important issue in modern astronomy is to clarify how perfectly the large-scale uniformity and homogeneity in matter distribution is maintained. Furthermore, researchers seek to investigate the properties of the seeds of large-scale structures (i.e., the initial matter fluctuations) that existed at the beginning of the universe. Thus, it is important to observe huge structures at various epochs (which translates to distances). The study of gaseous matter as well as galaxies is needed for an accurate and comprehensive understanding. This is because local superclusters are known to be rich in gas. Additionally, it is clear that there are many newborn galaxies in ancient (or distant) clusters. A detailed comparison between the spatial distributions of galaxies and gas during the early epochs of the universe is very important to understand process of galaxy formation from the dim (low light-emitting) clumps of gas in the early universe.

Astronomers take advantage of the fact that light from bright distant objects gets dimmed by foreground gas (giving an effect like a “shadow picture”) in order to investigate early, dim gas clouds. Since neutral hydrogen in the gas cloud absorbs and dims light from background objects at a certain wavelength, we can see characteristic absorption feature in the spectrum of the background object. In many previous observations, researchers used quasars (which are very bright and distant) as background light sources. Because bright quasars are very rare, opportunities for such observations are limited. This allows astronomers to get information about the gas that lies only along the line of sight between a single QSO and Earth in a wide survey area. It has long been the goal to obtain “multi-dimensional” information of gas (e.g., spatially resolve the gas clouds) rather than the “one-dimensional” view currently available. This requires a new approach.

**Expanding the View**

To widen their view of these objects in the early universe, Dr. Ken Mawatari at Osaka Sangyo University and his colleagues recently developed a scheme to analyze the spatial distribution of the neutral hydrogen gas using imaging data of galaxies of the distant epoch. There are two major advantages to this approach. First, instead of rare quasars, the team uses numerous normal galaxies as background light sources to investigate gas distribution at various places in the search area. Second, they use imaging data taken with the narrow-band filter on Suprime-cam. It is fine-tuned so that light with certain wavelengths can be transmitted, to capture evidence of absorption by the neutral hydrogen gas (the shadow picture effect). Compared with the traditional scheme of observations based on spectroscopy of quasars, this new method enables Mawatari and his collaborators to obtain wide-area gas distribution information relatively quickly.

The scientists applied their scheme to the Subaru Telescope Suprime-Cam imaging data taken in their previous large survey of galaxies. The fields investigated in this work include the SSA22 field, an ancestor of a supercluster of galaxies (proto-supercluster), where young galaxies are formed actively, in the universe 11.5 billion years ago in the early universe.

Schematic pictures of an analysis scheme of previous work (left) and a new method (right). In the previous approach, basically a single background light source (quasar) can be used in a searched area. On the other hand, with the new scheme, it is easier to spatially resolve the neutral hydrogen gas density by using many normal galaxies in a searched area as background light sources. In the new scheme, absorption strength by the neutral hydrogen gas is estimated by measuring how much flux of the background galaxies becomes dimmed in the narrow-band image, not by using spectrum. By combining this scheme with the wide-area imaging ability of the Subaru Telescope, Mawatari, et al. made the most-extensive map of neutral hydrogen gas ever created.

**New Maps of Neutral Hydrogen Distribution**

The scientists’ work resulted in very wide-area maps of the neutral hydrogen gas in the three fields studied. It appears that the neutral hydrogen gas absorption is significantly strong over the entire SSA22 proto-supercluster field compared with those in the normal fields (SXDS and GOODS-N). It is clearly confirmed that the proto-supercluster environment is rich in neutral hydrogen gas, which is the major building block of galaxies.

Sky distribution of the neutral hydrogen gas in the three fields studied in this work. While in the normal fields (SXDS and GOODS-N) the neutral hydrogen gas density is consistent with the average density in the entire universe at 11.5 billion years ago, the neutral hydrogen gas density is higher than the average over the entire SSA22 proto-supercluster field. Contours correspond to the galaxies’ number density. Bold, solid thin, and dashed contours mean the average, high density, and low density regions, respectively.

The team’s work also revealed that gas distribution in the proto-supercluster region does not align with the galaxies’ distribution perfectly. While the proto-supercluster is rich in both galaxies and gas, there is no local-scale dependency of gas amount correlated with the density of galaxies inside the proto-supercluster. This result may mean that the neutral hydrogen gas not only is associated with the individual galaxies but also spreads out diffusely across intergalactic space only within the proto-supercluster. Since the neutral hydrogen gas excess in the SSA22 field is detected over the entire searched area, this overdense gas structure is actually extended more than 160 million light-years. In the traditional view of structure formation, matter density fluctuation is thought to be smaller and large-scale high-density structure was rarer in the early universe. The discovery that a gas structure that extends across more than 160 million light-years (which is roughly same as present-day superclusters in scale) already existed in the universe 11.5 billion years ago is a surprising result of this study.

By investigating spatial distribution of the neutral hydrogen gas in a very large area, the scientists have provided a new window on the relation between gas and galaxies in the young universe. The SSA22 huge gas structure revealed by this work is considered a key object to test the standard theory of structure formation, and so further investigation is anticipated.

## Relic Radiation

Dark matter is so named because we cannot see it. So it’s ironic that much we have learned about dark matter has come from studying light, specifically, the cosmic microwave background (CMB). I often share with my students the story of its discovery, which paints a wonderful picture of how science works in practice and how we test scientific theory, although as an astroparticle physicist I do not study it directly.

The CMB story starts with Edwin Hubble, who made one of the most earth-shattering discoveries of the 20th century. In 1929 he found that the universe is expanding. After concluding that the “spiral nebulae” were “island universes” and not part of the Milky Way, Hubble measured their distances using Cepheid variable stars. Except for the nearby galaxies in our Local Group, all of the galaxies he observed were moving away from us, and the farthest galaxies were moving away the fastest.

The relationship of the velocity and distance for galaxies is linear and its slope is known as the Hubble constant, H_{0}. Hubble found its value to be about 500 km/s/Mpc, which means a galaxy one megaparsec from us will be observed to be receding at 500 km/s. The modern value is 69.32 +/- 0.80 km/s/Mpc. Hubble’s high value was due to errors in distances to galaxies. (Distances in astronomy are notoriously hard to measure.) This universal recession immediately suggested that the universe was nonstatic and evolving, and perhaps had a beginning.

Hubble’s discovery came at a time when a flurry of work was being done to model the universe at large using Einstein’s recently developed theory of general relativity. Einstein first favored a static, nonevolving model. However, Georges Lemaître, a Belgian scientist and Catholic priest, showed that an expanding universe was also a valid solution to Einstein’s field equations. Inspired by the phenomenon of radioactivity, Lemaître proposed that the universe as we see it began from the “decay” of a primeval atom. In his view cosmic remnants from this atom formed the seeds of stars, galaxies, and the other structures in the universe we see today. Lemaître viewed this as a cold process.

In the famous paper published in 1948, Ralph Alpher, Hans Bethe, and George Gamow proposed a model explaining the abundances of the elements that incorporated the expansion of the universe. The early universe, they argued, was hot and dense, and expanded from an initially ultradense state. They successfully calculated hydrogen and helium abundances however, they erroneously postulated that all heavier elements were created in the early universe through combining neutrons. We now understand that all elements heavier than lithium are created in the core of stars.

One of the most important predictions they made was too quickly forgotten: the initial hot, dense state of the universe should exhibit a leftover radiation field. In their theory, particles were created and annihilated in the early universe, and energy was transferred back and forth to a background of photons or light. Those frequent interactions meant that the universe could be modeled as a perfect blackbody, characterized by some temperature, *T*. As the universe expanded, this background of photons redshifted (i.e., lost energy). In essence, Gamow and his collaborators predicted the CMB and postulated that this background radiation should have a temperature today of about 5 K.

By the early 1960s cosmology had become a showdown between two competing theories. The big bang model gave the universe a problematically young age, two billion years. This age problem led Fred Hoyle, Hermann Bondi, and Thomas Gold to propose the steady-state theory, which explained Hubble’s expansion by proposing new physics and a static universe that continuously created new matter.

The two theories, big bang and steady state, gave very different predictions about the universe. In a way, the steady-state model was conceptually simpler it had fewer variable parameters and made more concrete predictions. One of these predictions was the distribution of radio sources at large distances. Measurements of radio sources seemed to disfavor the steady-state model, but the results were not conclusive at that time.

In 1964 astronomers Arno Penzias and Robert Wilson found the smoking gun that finally gave unequivocal evidence for the big bang model. While trying to calibrate a horn antenna at Bell Labs, developed to detect radio waves from satellites, they noticed excess noise in the sky corresponding to a uniform signal 100 times stronger than any background they had expected.

At first this signal frustrated them to no end. They went to extreme lengths, even removing bird droppings from the antenna, to determine the source of this background. After painstaking work, they found that the background was neither from the sun nor our own galaxy. It was extragalactic in nature, but its source remained mysterious.

Finally, when a friend pointed out the work of astronomers at Princeton University who were searching for the CMB, Penzias and Wilson realized what they had discovered. The two groups published joint articles in *The Astrophysical Journal* describing the discovery and interpreting it as the long-predicted cosmic microwave background radiation.

In 1989 NASA launched the Cosmic Background Explorer (COBE) satellite, which verified two fundamental properties of the CMB. The first was that the radiation is remarkably uniform (isotropic) across the sky hence the early universe was a nearly perfect blackbody. This discovery vindicated the use of statistical thermodynamics to describe the early universe.

But cosmologists were still puzzled by the uniform nature of the CMB. An extremely uniform CMB suggested an extremely uniform early universe. Why, then, is there structure today? Why isn’t the universe a dilute, uniform cloud of gas?

John C. Mather and George Smoot answered this question with COBE, which also revealed the second fundamental property of the CMB: although the CMB is remarkably isotropic, fluctuations (anisotropies) in temperature do exist. Some of the anisotropies discovered by COBE’s differential microwave radiometer (DMR) were due to our motion relative to the CMB frame and foregrounds, such as emissions from dust in the Milky Way. Once these anisotropies and other backgrounds were removed, fundamental anisotropies on the level of one part in 10 5 remained. In other words, one patch of the CMB sky differs in temperature from another at the fundamental level by only one 100,000th of a degree.

Those fundamental anisotropies were the seeds of early structure formation they allow us to figure out the composition and state of the early universe. For instance, the scale of these temperature fluctuations hints at the necessity of dark matter it is too small to allow ordinary matter time to coalesce into the structures we see today without the help of something like dark matter. The problem is time ordinary matter becomes charge neutral only at the epoch of recombination, and before that, due to electrostatic forces, matter cannot effectively clump into gravitational wells to begin forming structure. The COBE results showed a need for an electrically neutral form of matter that could jump-start the structure formation process well before recombination.

Mather and Smoot were awarded the Nobel Prize in Physics in 2006 for their measurements of the CMB.

After COBE, we have continued to learn a great deal more about the CMB thanks to the Wilkinson Microwave Anisotropy Probe (WMAP) and Planck satellite missions (among others). Experiments such as BICEP-2 (featured in this issue) are probing cosmic inflation shortly after the big bang using the polarization of the CMB. As a particle theorist, I continue to be amazed by the amount of information about the early universe that can be extracted from the cosmic microwave background.

**References**

Hubble, E., “A Relation Between Distance and Radial Velocity Among Extra-Galactic Nebulas,” *Proceedings of the National Academy of Sciences of the United States of America*, Volume 15, Issue 3, pp. 168–173 (1929).

Lemaître, Georges, “The Beginning of the World From the Point of View of Quantum Theory,” *Nature* 127, 706 (1931).

Alpher, R., Bethe, H., and Gamow, G., “The Origin of Chemical Elements,” *Phys. Rev.* 73 (7), 803–804 (1948).

Penzias, A. and Wilson, R., “A Measurement of Excess Antenna Temperature at 4080 Mc/s,” Astrophys. J. 142, p. 419–421 (1965).

Dicke, Peebles, Roll, and Wilkinson, “Cosmic Black-Body Radiation,” *Astrophys. J.* 142, 414–419 (1965).

**Reading list for more information**

Tegmark, Max, “Doppler Peaks and All That: CMB Anisotropies and What They Can Tell Us,” http://arxiv.org/abs/astro-ph/9511148.

Hu, W., Sugiyama, N., and Silk, J., “The Physics of Microwave Background Anisotropies,” *Nature* 386, 37–43 (1997).

## What is the hierarchy of structures in the Universe?

A computer simulation (more than 50 million light years across) of one possible scenario for the large-scale distribution of light sources in the universe. Credit: Andrew Pontzen and Fabio Governato.**What is the “hierarchy” of structures in the Universe? it seems that we have moon, then planet, then planetary system, then Galaxy, then maybe “several missing” in the hierarchy?, then universe.**

**How many “several missing” structures would there be? And would all of them just be considered clusters within clusters? Or is the “galaxy” structure the largest single independent organized type of structure in the universe and anything beyond that just one big disorganized mess?**

**Is there any organization beyond the galaxy structure, something containing galaxies, but again, not the entire universe and somewhat lesser than the entire universe? And if so, how many levels in the hierarchy are there before we get to “universe”?**

There are definitely structures larger than galaxies in the Universe, but it’s not clear exactly how many types or levels. The most obvious and uncontroversial class of structures beyond the galaxy is the galaxy group or cluster. These structures consist of tens to thousands of individual galaxies that are bound by gravity to one other and orbit a common center. That center is not defined by some huge central object, as the Sun defines the center of our solar system, but instead by the center of all the mass in the cluster, including dark matter. In fact, the dark matter dominates the mass budget of galaxy clusters, outweighing the normal matter by about a five-to-one ratio.

Beyond the scale of galaxy clusters, most astronomers agree that there are “superclusters,” consisting of several galaxy clusters and groups bound to one another. Our own galaxy and the galaxy group it is part of (the Local Group – a very creative name) is generally accepted to be part of the Virgo Supercluster, which may in turn be part of an even larger supercluster (the Laniakea Supercluster), though this finding is new and not fully accepted.

## Early Universe

### 12.5 Cosmic Background Radiation

Temperature plays a key role in the development of the early universe, because it determines which particle species were in existence. Equation (12.34) relates temperature at time *t* to the size of the universe at the same time, and we just found the size of the universe at time *t* for various dominances. The only thing left in finding the temperature as a function of time is determining T 0 , the present temperature of the universe.

In 1964, two radio astronomers, Arno Penzias and Robert Wilson, at Bell Laboratories were using a horn antenna, built earlier for communication via the *Echo* satellite, to measure the intensity of the radio waves coming from the Milky Way. The radio signals from all astronomical objects come in as “noise,” much like the statics picked up by radios during a thunderstorm. Distinguishing the signal noise from other spurious noises is not trivial, although it is much easier if the source is small, such as a star. In this case, one can switch the antenna beam back and forth between the source and the empty sky. If there is a detectable difference, it must be due to the source. The enormous size of our galaxy makes such a directional distinction difficult. In order to observe any signal from the galaxy, the antenna had better be as “noise free” as possible.

By a technique using liquid helium, Penzias and Wilson could reduce the expected spurious noise considerably. They started their observation in the spring of 1964 using a relatively short wavelength of 7.35 cm, where the radio noise from the Milky Way should have been negligible. To their surprise, they detected a strong signal. They changed the antenna direction the noise was still there. It appeared that the noise was coming from practically every direction. To make sure that the fault was not of the antenna, they dismantled the 20-foot horn, and discovered that some pigeons had nested in the antenna and deposited a “white dielectric material” there! After cleaning the mess and pointing the antenna to the sky in early 1965, they observed very little difference in the level of the noise. The noise did not want to quit.

Role of pigeons in the discovery of the big bang!

Puzzled by the persistent noise, Penzias contacted some colleagues, who eventually directed him to Princeton University. It turned out that a group of physicists there had been working on the formation of nuclei at the early universe. James Peebles, the theorist of the group, had argued that the observed structure of the visible universe, indicating a composition of about 99% hydrogen and helium, was a strong evidence for an intense radiation at the early universe. Peebles' calculation revealed that during the first few minutes of the evolution of the universe, the nuclear processes would take place at such a rapid pace that a large fraction of the nucleons would “cook” into the nuclei of heavy elements. The present absence of such elements must be the result of a mechanism that prevented their formation. The only candidate for such a prevention is a very dense and hot background radiation. Peebles estimated the present temperature of this radiation to be around 10 K.

Theoretical argument points to the existence of a background radiation in the universe.

In a subsequent meeting, the Princeton group and Penzias and Wilson decided to publish a pair of companion letters in the *Astrophysical Journal*, in which Penzias and Wilson would announce their observation, and the Princeton group would explain the cosmological implications. The radiation has come to be known as the **cosmic background radiation** (CBR).

If there is a radiation blanketing the universe, what kind of properties does it have? Is it a black body radiation? A black body radiator, by definition, contains radiation and matter in thermal equilibrium at some temperature *T*. In its early history, the universe consisted of protons, electrons, neutrons, and photons (plus neutrinos). Since photons interact very strongly with charged particles, these early photons were confined within the plasma of the charged particles and were in thermal equilibrium with them. Eventually, the electrons, protons, and neutrons combined to form neutral atoms of hydrogen and helium. Once the plasma was gone, the universe became transparent to photons and they were no longer in thermal equilibrium with matter they started to roam the universe freely.

When radiation was interacting with the hot plasma, its spectral energy density was described by the Planck radiation law u γ ( λ , T ) in (12.6) . Let *λ* and *T* be the wavelength and temperature of the radiation at time *t* when it was confined in the plasma. Let λ ′ and T ′ be its wavelength and temperature at time t ′ after it was decoupled from the plasma. Equations (12.32) and (12.34) give

Substituting in (12.6) , it is trivial to show that

So, except for a scale factor, u γ ( λ ′ , T ′ ) is identical to u γ ( λ , T ) . The radiation roaming the universe is indeed a black body radiation.

Penzias and Wilson could detect only a small portion of the curve characteristic of a black body radiator. As it reaches the Earth's surface, CBR loses most of its content to the atmosphere. Nevertheless, Penzias and Wilson could estimate the temperature of CBR to be about 3 K. To see the entire spectrum, and to determine the temperature more accurately, the antenna had to be lifted above the Earth's atmosphere. NASA's COsmic Background Explorer (COBE) satellite was launched on November 18, 1989.

Less than two months later, COBE had sent enough information that the investigators could construct the shape of the radiation curve. Was the curve that of a black body radiation as predicted by Equation (12.46) ? In the winter meeting of the American Astronomical Society held outside Washington, DC, on January 13, 1990, one of the principal investigators put up an image (reproduced in Figure 12.4 ) of the plotted data points on the screen. An eerie silence fell over the audience, which immediately turned into frenzied applause and a standing ovation. The data points fell exactly on the theoretical curve! The CBR *was* a black body radiator, and its temperature was 2.725 ± 0.001 undefined K .

CBR *is* a black body radiator, and its temperature is 2.725 K.

Figure 12.4 . The black body radiation curve as detected by COBE. The squares are the data points, and the solid curve is the BBR curve corresponding to a temperature of 2.725 K.

With the present temperature determined, I can now write (12.39) completely in terms of temperature

I can also give a relation between *T* and *t* for each stage of the early universe. From (12.34) , with a ( t 0 ) = 1 , I get T ( t ) = T 0 / a ( t ) and from (12.43) and (12.44) , I get

where T 0 = 2.725 undefined K , t H = 1 / H 0 = 4.59 × 10 17 s = 14.5 undefined Gyr , and *α* includes the contribution from all the particles (and their antiparticles) present in the universe at time *t*. A convenient relation that gives *t* in terms of temperature in the radiation-dominant universe is obtained by inserting all the numerical values in the equation for T α . Then, you can easily obtain

The temperature T eq that separates matter and radiation dominance is called the **equilibrium temperature** and is given roughly by setting the two densities equal. This temperature was reached far after all relativistic particles were gone except for photons and neutrinos. Therefore, I'll use α = 1.681 for ρ rel in ρ m = ρ rel . Then (12.30) , (12.35) , (12.38) , and our convention a ( t 0 ) = 1 lead to

Substituting the values of Ω m and Ω γ from (12.29) yields T eq ≈ 9300 undefined K and a eq ≈ 2.9 × 10 − 4 . Equation (12.49) now gives a *t* of about 65800 years as the time at which matter took over. Thus, for almost 66 millennia, radiation dominated the evolution of the universe. Although a minute fraction of its age, the radiation period was crucial in determining the fate of the universe, because during this period the elements from which every subsequent object was created were formed.

## Large-Scale Structure Formation: From the First Non-linear Objects to Massive Galaxy Clusters

The large-scale structure of the Universe formed from initially small perturbations in the cosmic density field, leading to galaxy clusters with up to 10 15 M_{⊙} at the present day. Here, we review the formation of structures in the Universe, considering the first primordial galaxies and the most massive galaxy clusters as extreme cases of structure formation where fundamental processes such as gravity, turbulence, cooling and feedback are particularly relevant. The first non-linear objects in the Universe formed in dark matter halos with 10 5 –10 8 M_{⊙} at redshifts 10–30, leading to the first stars and massive black holes. At later stages, larger scales became non-linear, leading to the formation of galaxy clusters, the most massive objects in the Universe. We describe here their formation via gravitational processes, including the self-similar scaling relations, as well as the observed deviations from such self-similarity and the related non-gravitational physics (cooling, stellar feedback, AGN). While on intermediate cluster scales the self-similar model is in good agreement with the observations, deviations from such self-similarity are apparent in the core regions, where numerical simulations do not reproduce the current observational results. The latter indicates that the interaction of different feedback processes may not be correctly accounted for in current simulations. Both in the most massive clusters of galaxies as well as during the formation of the first objects in the Universe, turbulent structures and shock waves appear to be common, suggesting them to be ubiquitous in the non-linear regime.

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