Report for the Modified Theories of Gravity

Inflationary Universe

The standard big bang theory is based on the cosmological principle, which states that the Universe looks the same to all observers. This statement put a constrain on the Universe that it must be homogeneous and isotropic, which can be described by the Friedmann-Lemaitre-Robertson–Walker (FLRW) metric of the form^[2]

in natural units where . Here, t is the time variable, and are spatial comoving (polar) coordinates. Given an object at rest on the coordinates remains the same with the expansion while the physical distances are related with comoving distances by

So, the function is called the scale factor, and basically, it measures the spatial expansion of the Universe. On the other hand, the constant parameter measures the spatial curvature, and with rescaling the coordinates, it takes one of the three discreet values corresponding to (hyperbolic) open, flat, and (spherical) closed Universes respectively.

After substituting the metric into the Einstein's equation,

we can get the equations of motion. For the simplest case, if we model the energy content of the universe as a homogeneous perfect fluid, the energy-momentum tensor, , takes the form where gives the energy density and is the pressure at given time . In that case, from and components of the Einstein's equation (where ), the EoM can be obtained as

where is defined as and called "Hubble parameter". In addition, from the energy conservation, namely the vanishing of the covariant derivative of the energy-momentum tensor, , we have the following relation, named fluid equation,

Together with and , these three equations are fundamental equations of modern cosmology. So, our aim is then trying to find the evolution of energy density and scale-factor with time for a given equation of state (EoS) for the content of the Universe.

For example, in flat geometry where , for matter, radiation and cosmological constant dominated Universes separately, whose EoS are , and , respectively, we have the following solutions for and as in the table below.

Moreover, it is useful to define two important parameters which are critical density and the density parameter . The former defines the necessary density for the Universe to be flat, and can be expressed as

while the latter is defined as the ratio of the energy density to the critical density at given time ,

Then, the Friedmann equation can be re-expressed in terms of the density parameter as

Note that for the mixed energy content, the density parameter can be separated as , and in fact, today's observation suggested that the Universe is composed of approximately of dark energy , of matter (including dark matter), and of radiation.

If we consider that the expansion is adiabatic, then from the first law of thermodynamics, , we have,

where is the entropy density so that the entropy is conserved.

From the Planck's energy distribution formula for the black body radiation, it can be shown that temperature and the expansion is related by , so that the Universe cools as it expands. Hence, it is reasonable to think that the Universe may have been arbitrarily hot and dense in its earliest stages. That means that in the earliest times where the temperature is very high, the Universe is dominated by radiation which includes both photons and ultra-relativistic all particles whose equation of state can be approximated by that of an ideal quantum gas of massless particles. In that case, the thermodynamical functions can be given by ^[1:1][3]

where is the value of Riemann zeta function at . The functions and can be expressed in terms of the number of bosonic spin degrees of freedom which are effectively massless at temperature and the corresponding number for fermions by

Hence, assuming that is not near any mass thresholds, the Friedmann equation can be re-expressed in terms of temperature as

where the function is defined by

I will use these functions when the problems with the hot Big Bang theory are introduced.

Like the cosmic microwave background radiation (CMB), detected by Arno Penzias and Robert Wilson in 1965, and the cosmic abundance of the light nuclear isotopes such as hydrogen, deuterium, helium-3 and helium-4, many other independent evidence support the big bang theory. However, despite the success of the old theory, it has some discrepancies deeply related to the initial conditions.

As examples, observations suggests that the Universe is very homogeneous and flat today. In fact, the level of anisotropies in CMB map is at and today's value of the density parameter is about . I will show how these observations causes problems.

From the Friedmann equation in terms of the density parameter, we can simply write,

Notice that the LHS of this equation measures how much the density parameter close to , and defines the flat geometry. The RHS, however, is always an increasing function of time both for matter and radiation.

So, is an unstable critical point, which means any deviation from will increase with time.

Today is certainly within an order of magnitude of one, and the age of the Universe is about order of sec, which suggests that in early times must be extremely small. For simplicity, just assuming radiation domination lasts up to today, gives

This result tells us that the Universe should have been way different if it did not start with an almost exact flat geometry.

Let us examine the problem using thermodynamical functions as in A. Guth's original paper^[1:2]. Since we consider the expansion is adiabatic, the entropy is conserved quantity, so its initial value can be determined or at least bounded by current observations. Assume roughly today's value of density is ; then, from Eq. , we have

for today. In general case ( ), this gives . By taking the present photon temperature as from the CMB spectrum, and using Eq. , it can be shown that the photon contribution to entropy is bounded by

Also, by taking the neutrinos and other effectively massless particles into account, the boundary to the total entropy can be given as,
so, using this and from Eq. , the value of the function is now

Therefore, using the density function , we can express the closeness to the flatness as,

where is the Planck's mass. For the very early times where the temperature is at , we have in the regime where the effective theory is grand unified theories (GUTs), and is typical of grand unified models. Therefore, we see that the Universe must be extremely close to the flatness at those times, namely

This is the flatness problem.

According to CMB observations, light coming from different regions of the sky seems to be at the same average temperature (about ), meaning that the Universe is very homogeneous. However, when you consider that the light have a limited distance to travel since its emission due to the finite speed of light, it came out that different regions of the sky would have never be in contact with each other in the past according to the evolution scenario in the hot big bang theory. This is a contradiction.

This problem can be shown quantitatively. A light pulse beginning at will have traveled a maximum physical distance

until the time , which is called the horizon. We will compare this horizon distance with the radius of the region at time t which will evolve into our observed region of the Universe. Now, due to the fact that is extremely small, if we ignore the term from Friedmann equation , the differential equation can be solved for temperature ,

where . For the minimal grand unified model, , so that .

From the conservation of entropy, notice that , so that . Therefore, Eq. becomes,

which gives the physical horizon distance.

In order the Universe to have a thermal equilibrium, this particle horizon should be in the same order with the radius of the observable Universe, ; however, from the conservation of entropy again, we can write,

where is the present entropy density and the current radius of observable Universe can be taken as . Hence, if we calculate the ratio of volumes,

Therefore, by taking and , we found

which means that there should be regions that are causally disconnected from each other in the observable Universe. This is called the horizon problem.

In order to avoid these problems, Alan Guth, in 1981, proposed an idea which is called the inflation. According to this, instead of an adiabatic expansion, suppose that the Universe has a short entropy generating period at initial where the entropy is greatly increased as

where and denote the present and initial values of entropy ( ), and is some large factor. This simple assumption solves the flatness and horizon problems.

Let us re-examine these problems with the assumption . The RHS of Eq. has a term now due to the function in . Then, the initial value of with could be of order unity,

if

Then, the flatness problem is solved.

As a solution to the horizon problem, due to assumption , the RHS of is multiplied by factor , so that if is sufficiently large, then the initial region which evolved into our observable Universe would have been smaller than the horizon distance at that time. Hence, the ratio of the volumes is multiplied by factor ,

Therefore, , if

In that case, the horizon problem is disappears. Notice that Eqns. and are approximately equal, since they both correspond roughly to of order unity.

In the next section, I will try to describe the mechanism behind this entropy generation.

In this section, I'll show how we can model the inflation, namely the exponential-like expansion at the very early stage. Starting with the Guth's model, we'll see different formalisms currently studied in the literature.

In the Guth's seminal paper^[1:3], he introduces a scenario which is capable of such a large entropy production. In short, he shows that if the equation of state for matter exhibits a first-order phase transition at some critical temperature , then during this phase transition we have a desired expansion to solve the big bang problems. He named named this scenario as the inflationary universe and describes it as the following.

Suppose the matter exhibits a first-order phase transition at a critical temperature, . As the universe cools to , bubbles of the low-temperature phase nucleate and grow. If the nucleation rate for this phase transition is rather low, the Universe will continue to cool as it expands.Therefore, it supercools to some temperature, which is many orders of magnitude below . When the phase transition finally takes place at temperature , the latent heat is released. Due to this heat, the universe is then reheated to some temperature which is comparable to as order. Assuming that the number of degrees of freedom for the two phases are comparable, the entropy density is then increased by a factor of roughly ,

Hence, the large factor can be read as,

which suggests that if the universe supercools by 28 or more orders of magnitude below the critical temperature, the horizon and flatness problems disappear.

Let us investigate the properties of this supercooling process a little bit further. As the temperature cools down to zero, , the system is cooling not toward the true vacuum, but rather toward some metastable false vacuum with an energy density ; which is necessarily higher than that of the true vacuum. In that case, the equation of density is now modified as,

which corresponds to a small modification of the Friedmann equation as well.

This equation has two solutions depending on the parameter .

If , where

then the expansion of the Universe has stopped at some temperature which is of , and then the Universe contracts again, which is the undesired scenario.

For case, only is physically plausible which correspond to the open universe. Once the temperature is low enough, the dominant term in RHS of Eq. is ; therefore, we have

where

Hence, from , we have,

meaning that the Universe is expanding exponentially, in a false vacuum state with energy density . The Hubble parameter is given by , or more precisely, approaches monotonically from above.

In literature, this scenario is described by scalar fields. Scalar fields could get caught in a local minimum of the potential, which in Guth’s work corresponded to a state with an unbroken grand unified symmetry. The inflation occurs due to a delayed first-order phase transition, in which a scalar field was initially trapped in a local minimum of some potential, and then leaked through the potential barrier and rolled toward a true minimum of the potential via tunnelling.

However, shortly after its publication, it is realized that this scenario has some problems.^[4] For example, the transition from “false vacuum” to the lower energy “true vacuum” could not have occurred everywhere simultaneously, but here and there in small bubbles of true vacuum, which rapidly expanded into the background of false vacuum, in which the scalar field would have been still trapped in its local minimum. Therefore, the latent heat released in the phase transition would have wound up in the bubble walls, leaving the interiors of the bubbles essentially empty, so that the only places where there would be energy that could grow into the present contents of the universe would be highly inhomogeneous and anisotropic.

Because of these problems, this idea is called "old inflation", today.

The solution to the problems of old inflation came with the “new inflation” theory in 1981-1982 by Linde^[5] and Coleman & Weinberg^[6]. In the new theory, inflation starts in an unstable state at the top of the effective potential, and then the field , called inflaton, slowly rolls down to the minimum potential. The density perturbations are produced during slow-roll and inversely proportional to which later become the seeds of galaxy formations. The consequences of the new inflationary theories turned out to depend on the "slow-roll" of the scalar field, rather than the process of bubble formation itself.

Now, let us examine the simple scalar field models to show how inflation occurs with this formalism. Suppose that we have a scalar field with a general action given by

where is the determinant of the metric, , the metric being flat for simplicity as .

Then, the solution to the Euler-Lagrange equations, the equation of motion can be derived as

We will be particularly interested in the homogeneous mode of the field , for which the gradient . In this case, the solution is reduced to

The stress-energy tensor for a scalar field is given by

where . Then, from , we have

We see that the de Sitter limit, occurs if the potential energy dominates the kinetic energy, namely . This limit is called slow-roll under which the universe expands quasi-exponentially. Therefore, We can rewrite the Friedmann and acceleration equations and as

where the parameter measures the accelerated expansion as , and the Planck mass is defined as . From these two equations it can be derived as

The de Sitter limit where is equivalent to , so we have

We will make an additional approximation that the friction term in the equation of motion is dominating, i.e. , so that the equation of motion for the scalar field is approximately

This approximation together with Eq. is called slow-roll approximation. Further, the condition can be expressed in terms of another parameter defined as

and together with

these are called slow-roll parameters. The slow-roll approximation holds when the slow-roll parameters are very small: . By using equations and , we can express these parameters in terms of the potential as

Moreover, we can define the number of e-folds N with the sign convention

so that increases as we go back in time. So, we can re-express the number of e-fold N in terms of the potential as

where denotes the field at the end of inflation.

Depending on the form of the potential, there are many models proposed. As examples, can be chosen as , etc. For given potential, the calculation of slow-roll parameters are trivial.

Lastly, I would like to note that there are two observable parameters depending on the slow-roll parameters, which are called the spectral index and the tensor-to-scalar ratio . These can be given as,

According to the recent observations, at CL [PLANCK 2018]^[7]:, and [combined analysis of PLANCK(2018) and BICEP2/Keck(2015)]^[8]. Therefore, model's free parameters (such as coupling constant ) can be fitted to some numerical values using these observable parameters.

In conclusion, we can picture the scalar field driven inflation as the following. At early times, the energy density of the universe is dominated by which is slowly evolving on a nearly constant potential that approximates a cosmological constant. Inflation ends as the potential steepens and the field begins to oscillate about its vacuum state at the minimum of the potential. Then, the energy of the inflation field must decay into the standard model of particles, and this process is called as reheating, which is beyond the scope of this discussion.

We saw that simply the vacuum energy ( ) with the EoS gives the desired exponential expansion; however, the mechanism behind the inflation cannot be a cosmological constant simply because a Universe dominated by the vacuum energy stays dominated by it for the infinite future, then a radiation dominated era will never be reached, so the “graceful exit” is not possible in this scenario. But, the desired expansion can be reached by a simple modification on the EoS as

The scalar field models can be reconstructed with a perfect fluid. There is also an increasing interest to imperfect fluids in the scientific community in order to explain the inflation better and to unify inflation with dark energy. For this purpose, two types of fluids have recently received attention, namely the Chaplygin gas^{[9][10][11][12]} and viscous fluids^{[13][14][15][16]}. For example the EoS of the modified Chaplygin gas can be given by

where are the free parameters needed to be determined by observations. The viscosity term can be added to the EoS as a function of Hubble parameter and its derivatives as

In these models the slow-roll and observable parameters can also be obtainable but here, the details of this formalism will not be discussed further.

Up to now, we have discussed how to realize the inflation by adding a hypothetical field or fluid to the RHS of Einstein's equation . The another route to make this happen is to modify the Einstein's relativity, namely the LHS of the equation. Since the inflation is thought to be happened at the very early stage of the Universe, the energy scale is very high, and at this scale the effective theory could be slightly different than the General Relativiy. It is possible that the quantum gravitational effects could play crutial role at this level, so people are investigating other theories to find out if the inflation is possible in those theories. Then, the general route to modify the Einstein's gravity is adding some geometric terms to the Einstein-Hilbert action. The Einstein-Hilbert action is given by

where is the Ricci scalar and is the matter action. The generalization of this action, on the other hand, can be done by replacing with a function of curvature terms as

Then, varying this action with respect to the metric gives the equations of motion similar to the Einstein's equation.

One of the examples of this kind is gravity where the function can be given by . The inflation in this theory is called the Starobinsky inflation, which is the earliest inflationary model proposed by Alexei Starobinsky in 1980^[17]. In this model, the modification to the action comes from some complicated renormalization procedures of conformal fields, but it result in graceful exit free inflation model with quasi-de Sitter expansion. There are so many model in this route and inflation in modified gravity theories is still an active research area.

In this report, I try to introduce the inflationary cosmology. To conclude, it can be said that the inflation is a paradigm to solve the problems of old big bang theory, some of which are presented in this report. Starting with the Guth's paper, it has taken a large attention in the science community and today, it is widely accepted idea but needs strong observational supports. Nevertheless, since the mechanism behind it is still not understood well, the inflation theory is one of the main branch in the studies of theoretical cosmology.