During the last decade, much progress have been done in the framework of so-called Loop Quantum Cosmology (LQC). Since the early works of e.g. Bojowald and Ashtekar, aiming at implementating Loop Quantum Gravity (LQG) ideas to the 'reduced case' of cosmology, physicists who embraced this idea are now able to derive non-singular quantum cosmological models and even to predict observational consequences for the B-mode anisotropies of the Cosmic Microwave background (CMB).
Rather a nice picture after ten years of work. But wait... Science is not as easy and some tensions and open questions in those (toy) models are still to be discussed and addressed.
Loop vs. String : a NON-SPECIALIST introduction
Building a quantum theory of the gravitational interaction is one of the big challenges in theoretical physics... and remains unsolved for more than forty years. The difficulties are technical : from a standard quantum field viewpoint, general relativity is not renormalizable and thus non-predictive (unless you consider that everything-you-can-compute=infinity is a prediction...); but also conceptual : the fundamental invariant of quantum mechanics, i.e. space-time, is the fundamental dynamical variable of general relativity. Different approaches to reconcile quantum mechanics and general relativity have been proposed and two of them are currently well developed (though their respective entire programs are not accomplished yet) : String Theory and Loop Quantum Gravity.
On the one hand, String Theory is essentially based on the usual perturbative quantization of Quantum Field Theory but replacing point-particles by fundamental strings. This theory is expected to unify all interactions as any particles could be found in the wide spectrum of strings. It is also a gravitational theory as a peculiar string vibration corresponds to a spin-2 particles interpreted as the graviton.
From a very naive point of view, this theory provides an renormalizable approach to describe gravitational interaction as gravitons propagated on some background space-time. However, in such a formulation, the theory is not diffeomorphism invariant : to quantize the theory, you need to assume a fixed background space-time. (Some correspondences between strong field and weak field formulation suggest that background independence should be present somewhere, though the author is not able to tell where.)
On the other hand, Loop Quantum Gravity aims at non-perturbatively quantize general relativity, keeping track of the fundamental symmetry of general relativity, i.e. diffeomorphism invariance. In other words, this theoretical building is quantized without assuming any background space-time by applying the Dirac quantization program to canonical variables which are diffeomorphism invariant.
The kinematic Hilbert space of the theory is given by the set of eigenvectors associated to some geometrical operators, like area or volume operators. Unlike sting theory, for which quantum states are found in a Fock space (particle interpretation), LQG quantum states can be viewed as quanta of space-time.
The last step of the LQG program, which is not accomplished yet, consists in solving the dynamic of the theory in order to compute e.g. transition amplitude between two different 'space-time' quantum states. (The spin-foam approach is one possible option to realize this dynamic.)
Quantum Gravity and Cosmology : a naive idea
The standard definition of 'cosmological physics' is "the study of the Universe as a physical entity". This is realized by considering the physical universe as the space-time containing all that is observed in the cosmos (an exercise for the reader: Define cosmos?). Being a space-time, the dynamic of our Universe, i.e. its geometry, is described by general relativity. And if for any reason, our Universe appears to be, in the past or in the future, quantum-sized, the problem of quantum gravity pops up in the field of cosmology.
Under the hypothesis that the cosmological principle holds (the Universe is homogeneous and isotropic), the Universe is like the stack of three-dimensional elastic sheet, denoted space, which evolve as a function of a fourth dimension, denoted cosmic time. Those elastic sheets can either be stretched : the Universe is expanding; or can shrink : the Universe is contracting. Thanks to E. Hubble, we know from observation that our Universe is expanding. But was it always the case?
Using general relativity and standard particle physics, it is shown that the Universe cannot transit from a contracting phase to an expanding one. Because of this, if the Universe is currently expanding, it was also expanding in the past. In other words, by going back in time, the Universe is contracting. Does this mean that our Universe can be infinitely extended in the past?
Using general relativity and standard particle physics, it is shown that by going back in time, the Universe contracts to a singularity, a point in space-time for which all physical quantities (density, temperature, space-time curvature etc.) become infinite valued. At this point, space and time ends and the theory is no more predictive.
But such a conclusion holds if one assumes that general relativity is a valid framework close to the so-called Big-Bang singularity. However, close to the singularity, the Universe precisely becomes quantum sized and the use of quantum gravity is then mandatory.
The basic picture is therefore the following : before the Planck time, the energy density of the universe is of the order of the Planck energy and nothing can be said except in the language of quantum gravity. Let's call this 'the quantum universe'.
Then, it is widely believed that the Universe went through a phase of accelerated expansion called inflation. During this epoch, gravity is classical but matter and radiation fields are quantum. Though not unambiguously proven by observation, the inflationary phase solves many problems in cosmology. For example, this is during this phase that the primordial seeds of galaxies and large scale structures (local over- and under-densities), and primordial gravity waves are produced. And those tiny inhomogeneities leaves a clear footprint on the temperature and polarization CMB anisotropies as well as on the distribution of galaxies. At the end of inflation, the Universe should went through a phase of reheating during which (more or less) standard particles via the decay of the field at the origin of inflation. Though inflation solves many problems, it is important to keep in mind that initiated a long enough inflation is rather complicated and requires non-standard physics invoking e.g. new quantum fields or even quantum gravity. Let's call this 'the primordial universe'.
Then the Universe keeps expanding and cools down. Hundred seconds after the initial singularity, the first nuclei form via nuclear fusion. The universe is filled with a homogeneous, but slightly perturbed, ionized plasma made of Dark Matter (living rather independently) and baryons coupled to electrons, themselves coupled to photons. 380,000 years after the initial singularity, electrons are captured by protons : the first atoms form in the Universe and photons abruptly stop to interact with electrons. Light can then propagate freely, corresponding to the release of the CMB photons in the Universe. And because the Universe is filled by some tiny primordial density perturbations and primordial gravity waves, these leave some imprint on the CMB properties. Let's call this 'the early Universe'.
Then, because of gravitational collapse around local over density, galaxies, large scale structure, stars etc. start to form in the Universe. Dark Energy starts to dominate the total energy budget in the Universe (precisely at the time when the Universe becomes structured). Let's call this 'the Universe studied by other cosmologists but still very interesting to me'.
Elaborating on this, it is easily understood that primordial cosmology is a suitable laboratory to 'experiment' quantum gravity proposals.
On the one hand, we have this well-posed theoretical problem, namely how to quantize gravitation. And searching for very high energetic phenomena, close to the Planck energy scale, is mandatory to observationally probe quantum gravity.
On the other hand, we have the inflationary paradigm, a very high energetic states of Universe which leads to some observational footprints on the CMB anisotropies.
By gluing the quantum Universe to the primordial Universe, it is in principle possible to analyze how quantum gravity influences, or even trig on, inflation and how this translates on the CMB anisotropies.
In the following of this post, we will mainly discuss how this idea is realized in the context of Loop Quantum Cosmology.
Loop Quantum Cosmology : the background
The usual framework adopted in cosmology is to disentangle the homogeneous background from the tiny inhomogeneities, like primordial gravity waves, propagating on it. How such a dichotomy is realized (if possible) is beyond the scope of this post and the results presented below assumes that the largest scale corresponding to the homogeneous background is not affected by the smallest scales corresponding to the tiny inhomogeneities of the Universe. (More precisely, the background evolution is only affected by the first momentum of the statistics of inhomogeneities, which is zero by definition of the inhomogeneities.)
LQC corresponds to the implementation of LQG idea to the symmetry reduced case of Friedman-Lemaitre-Robertson-Walker metric, the line element describing an isotropic and homogeneous space-time :
(A spatially flat Universe is considered here.) The dynamic of cosmology can be first re-written in the Hamiltonian framework proposed by Ashtekar, the two dynamical variables being the square of the scale factor and its time-derivative. At a classical level, the evolution of the Universe is, obviously, exactly the same adopting the Ashtekar formalism or the standard approach.
Then, quantization of the theory is done using a new set of dynamical variables : the holonomies of the Ashtekar connexion and the fluxes of the densitized triad. (Those new variables are easily promoted to well-defined operators and ensure background independence.) In the Hilbert space emerging from the quantization procedure, some semi-classical states are identified. Those states corresponds to wave packet which are strongly localized around the classical trajectory. However, the trajectory drawn by those states becomes rather peculiar close to the Planck scale allowing a contracting Universe to bounce back and then to enter in an expanding phase. Those trajectories are indeed described by the following effective, modified Friedman equation
the quantum analog of the usual Friedman equation
The Friedman equation relates the expansion of the Universe (the left hand side) to its energetic content (the right hand side). Because of the additional term in the right hand side of the modified Friedman equation, it is now possible to have a Universe transiting from a contracting phase to an expanding phase, this expanding phase being interpreting as our expanding Universe while the bounce replaces the Big-Bang singularity.
In the framework of LQC, it therefore appears possible to replace the singular Big-Bang Universe by a non-singular Big-Bounce Universe.
In the above, the content of the Universe was not specified. An interesting model recently developed consists in considering that the Universe is filled with a massive scalar field, whose evolution is given by the Klein-Gordon equation
We can now consider that during the contracting phase, the scalar field is at rest. However, because of quantum fluctuations, this field may have a slight amount of momentum. Moreover, during the contracting phase, the second term in the Klein-Gordon equation acts as anti-friction. As a consequence, the scalar fields will climb up its massive potential during contraction. Then after the quantum bounce, the field may be very high in its potential which are the appropriate conditions for a phase of inflation to start at the beginning of the expanding phase. The evolution of the scalar field and the scale factor are displayed in the two following figures (taken from Mielczarek et al., Phys. Rev. D 81 104049 (2010)) :
In other words, the contraction plus the quantum bounce set the Universe in the appropriate conditions for a phase of inflation to start.
However, inflation should be long enough to solve the horizon and flatness problems. And a long enough inflation means 60 e-folds (that is the size of the universe is multiplied by at least exp(60) during inflation). In the above described model, a long enough inflation is reached if few percent of the total energy of the scalar field at the time of the bounce is in the form of potential energy. A more detailed analysis has been performed by Ashtekar and Sloan who studied the probability for a 60 e-folds inflation to appear in this kind of model. They concluded that in the LQC framework, a long enough inflation is highly probable, which is not the case in more standard inflationary models (see Ashtekar & Sloan, arXiv:0912.4093v2 [gr-qc]).
To conclude, LQC offers the possibility to solve first for the initial singularity and second for the apparent lack of naturality of inflation.
Moreover, it appears that a 60 e-folds inflationary phase is usually met and the LQC induced inflation is therefore long enough to solve the different problems of non-inflationary cosmological models.
Finally, because cosmological perturbations experienced first the contracting phase and the quantum bounce before being amplified during inflation, one may expect the statistical properties of the primordial perturbations to differ from the standard inflationary prediction; offering here a potential way to test the model with CMB anisotropies.
We underline that the model presented above incorporates LQC correction from the use of holonomies only. In the framework of LQC, a second class of quantum corrections may arise from the fact that volume operators are quantized. The nature (and existence) of such corrections are still debated and we will restrict to models with holonomy corrections in this post.
Loop Quantum Cosmology : the inhomogeneities
The amount of inhomogeneities produced in this {contracting+quantum bounce+inflation} Universe is in principle obtained by studying the perturbed FLRW metric and Klein-Gordon equation, both re-casted in the language of LQC to incorporate quantum corrections.
Two types of cosmological perturbations are expected to be produced : i) scalar perturbations (more or less ta peculiar combination of density and Newton potential perturbations) and ii) tensor perturbations, i.e. primordial gravity waves. However, the full LQC quantization of the perturbed FLRW metric has not been performed yet and only anomaly-free effective approaches has been developed for the specific case of tensor perturbations by Bojowald & Hossain, Phys. Rev. D 77 023508 (2008).
With such an effective approach, it is then possible to derive equation of motion of primordial gravity waves in the LQC Universe taking into account quantum correction. Basically, the propagation of gravity waves, as compared to the standard result, is modified in two ways : i) the evolution of the background involves the contraction and the bounce and ii) the dispersion relation of gravity waves is modified because of the quantum nature of the background.
Thanks to numerical resolutions of the equations describing the evolution of both the background and the tensor perturbations, incorporating first order LQC corrections, it was shown in Mielczarek et al., Phys. Rev. D 81 104049 (2010) that the primordial power spectrum for gravity waves at the end of inflation exhibits the following properties (we remind that in the standard inflationary scenario, this power spectrum is a power-law as a function of the spatial frequency of the waves) :
- at large scale, the power spectrum is suppressed as compared to the standard prediction;
- at a critical length scale, the power spectrum shows a rather high bump;
- after this critical length scale, the spectrum shows some damped oscillations;
- at small length scale, the power spectrum is a power law which coincide with the standard prediction.
This results can then be used as initial condition for the angular power spectrum of the B-mode CMB anisotropies, which is the (unfortunaltely not yet detected) observational quantities. The discrepancies between the LQC prediction and the stand prediction are displayed below :
The interpretation of such distortions is simple. First of all, the large length scale suppression translate into a large angular scale (i.e. small multipoles) suppression while the bump at a critical length scale roughly translates into a bump at some critical mutlipole. The amplitude of the bump in the B-mode is directly proportional to the amplitude of the bump in the primordial tensor power spectrum. The multipole location of the bump in the B-mode power spectrum strongly depends on the value of the critical length scale. One can essentially identify two regimes. First, if the critical length scale is smaller than the Hubble scale today (roughly 500 Mpc), the bump shows up at multipoles greater than ~10. Second, if this critical length scale is greater than the Hubble scale, then the bump in the B-mode shows up at multipoles lower than ~10, and the suppresion cannot be observed.
From an observational viewpoint, the height of the bump is directly linked to the mass of the inflaton field while its location is linked to the percentage of potential energy in the inflaton field at the bounce. Obviously, estimating those parameters from the B-mode power spectrum recovery is spoilt by different degeneracies (in particular with the properties of reheating). Nevertheless, part of those degeneracies could be broken by also measuring the normalization and spectral index of the primordial power spectrum at small length scale.
The end of the story
The model is clearly appealing. It solves the old problem of the initial singularity. It naturally leads to a long enough phase of inflaton. And it appears possible to test it by observing the B-mode anisotropies of the CMB. Nevertheless, this type of models are not complete yet and some open questions are still to be discussed.
First of all, the case of scalar perturbations has to be considered. The problem seems more technical than conceptual and consists in finding how to implement the first order quantum correction without anomaly to the dynamic of those perturbations.
However, computing the power spectrum of scalar perturbations is a crucial step allowing us to test the model with current measurements of the Temperature and E-mode anistropies of the CMB. (A new model should, at least, be in agreement with current data...)
Secondly, it has been tested that the effective, modified Friedman equation is a good approximation of the full evolution of the wave packet for massless scalar field. If the scalar field exhibits some auto-interaction potential, the modified Friedman equation holds if the amount of potential energy at the bounce is small. Though a long enough inflation is realized with such a small amount of potential energy, it is not guaranteed that this is still the case if a substantial amount of energy at the bounce is contained in the potential energy.
Finally, and more importantly, disentangling the homogeneous background from the influence of inhomogeneities may be a spurious hypothesis. Though it appears as a rather fair assumption in the classical regime (and successfully used in standard cosmology), the inhomogeneities are highly concentrated at the bounce and higher momentum of their statistics may play a crucial role on the background.
Considering that inhomogeneities affect the background can for example be realized by the fact that the full quantum state describing our Universe with perturbations cannot be separated into a homogeneous part and inhomogeneous part. And this may change the basic picture presented above.
Important questions are hidden behind this problem. For example, how LQC can emerge from LQG using a complete top-down viewpoint. Recently, a quantum model of the Universe incorporating few inhomogeneous degrees of freedom has been built using Loop Quantum Gravity (see Battisti, Marciano & Rovelli, Phys. Rev. D 81 064019 (2010)). Another important related question to this is also : is the cosmological principle still valid at a quantum and/or quantum gravity level and, if the answer is yes, how it can be implemented?
Apologies...
In this brief introduction to quantum models of the Universe inspired by Loop Quantum Gravity, some choices have been made. This long post therefore does not give justice to all the people working in the field. Sorry in advance...
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