9 March 2019
9 Mar 2019
21 min

# 1 Introduction

Some say the world ends in fire, the 2011 Nobel laureates say in ice. Their findings prove an extraordinary hypothesis that our universe is subject to an ever-accelerating expansion [1,2]. Based on Einstein’s first conjecture that there must be an antigravity constant to keep physical bodies in the universe from collapsing, cosmologists have found that a colossal portion of the cosmos is made up of an unknown energy. The observations sparked many new theories attempting to explain this so-called dark energy. A frontrunner from the very beginning is a cosmological constant model; however, this thesis will also try to shed light on an alternative proposition, in particular, a modified gravity model.

# 2 Distance

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## 2.1 Supernovae Type Ia

Distance measurement in the universe is tricky. The farther away the object, the more sophisticated methods are used. For galaxies near the Milky Way, Cepheid variables are excellent distance indicators. There are plenty of these bright stars available and to measure a distant object, one creates a cosmic distance ladder from one Cepheid to the object of interest. As observable objects are further and further apart, the Cepheid variable method becomes too imprecise as the brightness fades with the square distance [19]. An obvious solution: find a brighter star.

For a long time, the brightest galaxies in the universe were used to measure distances of far away objects. However, galaxies evolve and so do their absolute luminosity. It has been found that type Ia supernovae have stable absolute luminosities [20].

Supernovae of type Ia (SNe Ia) are thought to arise from a thermonuclear explosion of an unstable white dwarf, unlike other types of supernovae, which are the result of the explosion of a main sequence star. A white dwarf that exists in a binary system (two stars orbiting each other) is said to accrete gas from its partner star until it has passed the Chandrasekhar limit of $1.44M_\odot$ beyond which the pressure of electron degeneracy of the white dwarf can no longer keep it from collapsing. SNe Ia are different from other supernovae in that they do not have hydrogen lines but a strong silicon absorption line at 615nm in its spectrum. The collapse results in an increase in temperature and density, which compresses carbon and oxygen to $^{56}Ni$, which then again triggers two radioactive decays and a thermonuclear explosion that leads to a bright supernova [3].

Due to approximately the same mass at explosion, SNe Ia are expected to produce the same absolute luminosity, yet ultimately it depends on the amount of nickel present in the white dwarf’s core. Explicitly, the more nickel, the more luminous the supernova.

A higher temperature in gas means a higher opacity that traps the (light) energy from escaping. Therefore, the brighter the supernova, the slower its brightness decline, in other words, the wider its light curve. Phillips [4] observed a tight correlation between peak magnitude and light curve stretch of SNe Ia. Explicitly, he found that one can normalize the light curves of many supernovae and find only a minimal variation in the peak magnitudes.

SNe Ia, like Cepheids for shorter distances, are “standardizable candles” due to their consistent level of brightness. Standard candles have a known absolute source strength (or luminosity, $\mathcal{L}$) that makes it possible to count the photons going through a particular surface area. SNe Ia radiate spherically and therefore the observed apparent luminosity (or bolometric flux, $\mathcal{F}$) is related to the inverse square law of the distance to the object.

$$\mathcal{F}=\frac{\mathcal{L}}{4 \pi d_L^2}$$

This distance is called luminosity distance and can be related to redshift $z$ through the Hubble distance $H(z)$ in the Friedmann equation. Unless otherwise specified, we set the speed of light constant to one ($c=1$) throughout this thesis.

$$d_L(z)=c\cdot(1+z)\int_0^z\frac{d\tilde{z}}{H(\tilde{z})}$$

Figure 1. Distance luminosities over a range of redshifts for different cosmic consistency as defined by the normalized energy density $\Omega_m$.

The concept of magnitude was formalized by Hipparchus in 120 BC, and three centuries later by Claudius Ptolemy. He introduced categories for the apparent brightness of stars on a scale from one to six “magnitudes”. Later this notion was quantized to relate apparent luminosity to a factor of $100^{-m/5}$. Specifically, two objects with a magnitude difference $∆m$ have an apparent luminosity of $10^{-∆m/2.5}$ [20]. The brighter the star, the lower its magnitude and a difference of five magnitudes equals a factor 100 in luminosity.

$$\frac{d_L}{d_{L,0}}=10^{(m-m_0)/5}$$

For a standardized model, we take the reference magnitude $m_0$ as absolute Magnitude of a known object, i.e. the sun, which is produced at the reference distance luminosity $d_{L,0}=10pc$. Magnitudes we usually want to observe are of objects at cosmological distance (i.e. $1Mpc$).

\begin{aligned}m&=m_0+5\log_{10}({\frac{d_L}{d_{L,0}}})\cr&=M+5\log_{10}({\frac{d_L}{1Mpc}})+5\log_{10}({\frac{10^6pc}{10pc}})\cr&=M+5\log_{10}d_L+25\end{aligned}

Figure 2. Relative magnitude for various $\Omega_m$ over a range of redshift values, $𝑧\in[0,2]$

## 2.2 ΛCDM Model

There are two sides in Einstein’s equation of general relativity – the gravity and the matter (energy-momentum) terms. Before discovering that the universe is in fact expanding, Einstein thought that the universe was static ($\dot{a} = 0$). With this in mind he designed an extension to his equation of general relativity that has an energy density equal to its negative pressure and therefore results in the addition of a constant.

$$G_{\mu\nu} = 8\pi G(T_{\mu\nu} + \rho_\Lambda g_{\mu\nu})$$ $$-p_\Lambda = \rho_\Lambda = \frac{\Lambda}{8\pi G}$$

The first component of the energy-momentum tensor ($T_{00}$) corresponds to the energy density ($\rho_m$), which together with the Friedmann-Robertson-Walker (FRW) metric leads to the Friedmann equation that describes the distance evolution of our universe.

The FRW metric describes a line element in a four-dimensional homogenous, isotropic universe.

$$ds^2 = -dt^2 + a(t)^2(dr^2+S_k^2(r)d\Omega^2)$$ $$S_k(r) = \begin{cases} \sin r &k = 1 \cr r &k=0 \cr \sinh r &k=-1 \end{cases}$$

$a(t)$ is the expansion factor and is related to the inverse redshift, $a=(1+z)^{-1}$. $S_k(r)$ describes the curvature of the universe; this thesis we will focus solely on a (spatially) flat universe, i.e. $k=0$.

Under the conversation of energy, it is possible to assume the densities for a non-relativistic matter term and a dark energy term with the following properties $(H=\frac{\dot{a}}{a})$.

\begin{aligned} \dot{\rho_m} + 3H\rho_m &=0 \cr \dot{\rho_\Lambda} + 3H(\rho_\Lambda + p_\Lambda) &= 0 \implies \rho_\Lambda = const.\end{aligned}

$$H^2 = \frac{8\pi G}{3}(\rho_m + \rho_\Lambda) \tag{1}$$

Using the current composition of the universe we can relate the Hubble distance and the redshift.

\begin{aligned}H^2(z) &= \frac{8\pi G}{3}(\rho_{m,0}(1+z)^3 + \rho_\Lambda) \cr &= H_0^2\frac{8\pi G}{3H_0^2}(\rho_{m,0}(1+z)^3 + \rho_\Lambda) \cr &= H_0^2[\Omega_m(1+z)^3 + \Omega_\Lambda] \tag{2} \end{aligned}

Furthermore, it is common to normalize energy densities as follows:

$$\Omega_i = \frac{8\pi G}{3H_0^2}\rho_i = \frac{\rho_i}{\rho_crit}$$

So that, at present epoch ($z=0$) the universe is only made up of non-relativistic matter and dark energy: $1 = \Omega_m + \Omega_\Lambda$.

### 2.2.1 Shortcomings

There are two major problems with the cosmological constant Cold Dark Matter (ΛCDM) model. The first is referred to as the fine-tuning dilemma. From the assumption that the cosmological constant is a vacuum energy, where the ground state energy (assuming it behaves like a harmonic oscillator) is $\hbar w / 2$, we can infer the theoretical vacuum energy density by integrating up to a certain cutoff value, like the Planck scale $M_pl$. The theoretical value of this vacuum energy is $(10_{18}GeV)^4 \simeq 2\times10^{110}erg/cm^3$ whereas the commonly observed value is $(10_{-12}GeV)^4 \simeq 2\times10^{-10}erg/cm^3$ making it consistently an enormous difference of 120 magnitudes [5].

The second problem is concerned with the order of magnitude of the dark energy density. Not only is it small, but it is about the same size as our universe’s current mass density ($\Omega_m \sim \Omega_\Lambda$). This implies that our universe must have just started to accelerate its expansion. [5] For the lack of a better comprehension, this coincidence problem is written off with the anthropic principle. That is, we, observers, are only allowed to observe certain states.

Even though we can only explain limited features of this mysterious dark energy that is believed to be the driving force behind the acceleration of our universe, Einstein has been all but wrong to introduce $\Lambda$. The ΛCDM model still explains the universe, as we know it, to the best of our knowledge. There are many theories to modify either the gravity or matter term of the general relativity equation, and maybe with the next round of high-precision observations one of these alternative theories will yield a significant advancement.

We will now take a look at one of these alternative theories, namely the Dvali- Gabadadze-Porrati (DGP) model. This model considers changes to the gravity term of the Einstein equation eliminating the need for a dark energy variable.

# 3 DGP Cosmologies

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## 3.1 A New Approach

The DGP model postulates that gravity is not understood correctly. It is also called a DGP braneworld theory because it embeds a brane – our four-dimensional (4D) world (including the temporal dimension) – in an infinite-dimensional large Minkowski bulk [6]. However, for simplicity we will only look at one extra dimension (5D) in the assessment of the model action and thereafter. Matter and all forces reside on the 4D brane, while gravity is allowed to crossover to and float freely on the extra dimension of the 5D bulk.

The action of the DGP model is given by

$$S = M_(5)^3 \int d^3x \sqrt{-g_{(5)}} R_{(5)} + M_{pl}^2 \int d^4x \sqrt{-g}R + \int d^4x \sqrt{-g} L_m$$

Properties of the 5D bulk are denoted by the subscript (5), while $R$ and $g_{\mu\nu}$ are the Ricci scalar and the metric induced on the 4D brane with $g$ as its determinant, respectively. The DGP model action is simply a sum of a 5D action, a 4D action, and a matter component in 4D.

Dvali, Gabadadze, and Porrati derived a form of the gravitational potential for the 5D bulk and found that at short distances the potential has a 4D Newtonian $1/r$ dependence, whereas at large distances the potential scales in accordance with inverse squared law ($\varpropto 1/r^2$). This dynamic arises from the clash of the 4D Ricci scalar and the 5D Einstein-Hilbert action and is quantized by the crossover scale factor [7]:

$$r_c = \frac{M_{pl}^2}{2M_{(5)}^3}$$

The dilution of gravity at extremely large scales is interpreted by DGP as gravity crossing over from the 4D brane and leaking into the 5D bulk. This understanding lays a solid foundation for an accelerated universe at cosmic distances. At these scales, the gravitational drain diminishes the effect of gravitating matter, hence speeding up the expansion and eventually reaching the de Sitter universe.

For a practical application, including the modified gravity term in Einstein’s field equation, we can construct a new cosmology. This all boils down to a set of modified Friedmann equations

$$H^2 \pm \frac{H}{r_c} = \frac{8\pi G}{3} \rho_m \tag{4}$$

$$\dot{\rho_m} + 3H(\rho_m + p_m) = 0 \tag{5}$$

The first equation (4) is modified by introducing a $\pm H/r$ term, which arises from a square root treatment when solving the modified equation of general relativity.
In the early universe, when the first term ($H^2$) dominates (4), the equation can be described by the conventional 4D Friedmann equation (1). In the later stage, the second term dominates and a split of paths between a shift to the 5D FLRW phase and to a self-inflationary phase is manifest [6].

The two signs both represent a different cosmological regime. The positive sign refers to an existing theory with the Friedmann-Lemaître-Robertson-Walker (FLRW) phase, which at short distances expands with the energy density and later takes on the nature of the 5D FLRW state. In the case of a negative sign, the early universe is described by the conventional 4D Friedmann equation $H^{-1} \ll r_c$, however as the Hubble distance gets large so that matter in the universe is extremely reduced and eventually lacking thereof entirely, the second term dominates and the expansion approaches a constant value, $H(\infty) \sim 1/r_c$ [8]. This self-inflationary state is called the de Sitter universe, dominated by a single constant expansion factor, which outside of modified gravity theories – for the lack of a better understanding – is simply called the cosmological constant ($\Lambda$).
In addition to a flat universe, for the rest of the discussion, we will focus on the lower sign that leads to the self-accelerating phase.

The energy conservation equation (5) is unchanged from the conventional cosmology theory, thus the modified Friedmann equation can be written as a construct of normalized densities and redshift.

\begin{aligned} H^2 - \frac{H}{r_c} &= \frac{8\pi G}{3} \rho_m \cr &= H_0^2 \Omega_m (1+z)^3 \end{aligned}

for $z = 0$:

$$H_0^2 - \frac{H_0}{r_c} = H_0^2 \Omega_m$$

$$\implies 1- \Omega_m = \frac{1}{H_0r_c} \eqqcolon \Omega_{r_c} \tag{6}$$

$$H^2(z) = H_0^2 [\Omega_m (1+z)^3 + \frac{H}{H_0} \Omega_{r_c}] \tag{7}$$

The rest of the standard cosmology is replicated perfectly by the DGP model. Especially for a flat universe, the DGP-Friedmann equation (7) and the conventional Friedmann equation (1) are very similar and parallels between $\Omega_{r_c}$ and $\Omega_\Lambda$ become apparent. From (6),

$$1 = \Omega_m + \Omega_{r_c}$$

Although the DGP model is a very elegant and attractive alternative to the standard model, observations have not been in favor of its validity. Zhang et al [10] has compared five modified gravity models – DGP braneworld included – to the ΛCDM model with cosmological data from Supernova Legacy Survey (SNLS) SNe Ia probes, seven-year WMAP CMB anisotropy data, BAO data, and latest Hubble constant measurements. The authors calculate the combined $\chi^2$ of each independent cosmological data source for redshifts $z \in [0.15, 3.8]$ and compared the result to ΛCDM. To account for relative goodness of fit, the authors also employ Akaike (AIC) and Bayesian information criterion (BIC).

Since we are only interested how well the DGP model performs, we highlight a direct comparison between the growth factor $f(z)$ and the results for cumulative $\chi^2$.

Figure 3. Left: The growth factor $f(z)$ according to ΛCDM and DGP models. SNe Ia+CMB+BAO+H0 observations dictate the best-fit and $2\sigma$ range. Right: ΔBIC ≥ 10 is strong evidence against the DGP model. ΛCDM is the preferred choice here. Taken from [10].

The DGP braneworld is very far off from matching the observational data. In a way, we are lucky that previous research [11] has not been any more favorable toward the DGP model. In the meantime, it is plausible that we are just looking at a crude version of the DGP model altogether.

## 3.2 The Extension to the Modification

In light of this catastrophic model-observation conundrum, Dvali and Turner [12] attempt to salvage the DGP model by proposing an extended version of the braneworld. The corrections to the conventional Friedmann equation are now given by a parameterized factor $H^\alpha$, with the new formula:

$$H^2 - \frac{H^\alpha}{r_c^{2-\alpha}} = \frac{8\pi G}{3} \rho_m$$

The motivation behind this further modification is that if at larges distances the gravity changes, but the observations do not fit the current model, maybe we ought to take smaller steps in between these phase changes. At small scales, the new DGP model still does a great job of replicating the standard Friedmann model, and thus gives an upper bound on the value of $\alpha$. In the case, when $\alpha = 2$, no matter can exists in the universe according to this model ($H^2 - H^2 = 0 = \frac{8\pi G}{3} \rho_m$), hence the first constraint is $\alpha \le 2$. Especially, the process of the origin of our physical universe – the big bang nucleosynthesis – constrains the choice to $\alpha \le 1.95$. Furthermore, the matter-dominant epoch cannot have sustained its longevity if $\alpha \gt 1$ [12].

From the extended DGP (αDGP) model, we can easily infer the flat DGP model ($\alpha = 1$) and the ΛCDM model ($\alpha = 0$), where for the latter case, $\Lambda$ can be constructed from the constant crossover rate:

$$\frac{H}{r_c} = \frac{8\pi G}{3} \rho_\Lambda = \frac{\Lambda}{3}$$

Thus, in a way, αDGP is a generalized formula, combining a modified gravity model and a cosmological constant model, to describe the universe.

As with the flat DGP cosmology, we can construct a αDGP cosmology with $\Omega_{r_c}$ and $\Omega_m$. In the results section, we work with a αDGP model, which only relies on $\Omega_m$, $\alpha$, and $z$.

$$\frac{H^2}{H_0^2} - \Omega_{r_c} \frac{H_\alpha}{H_0^\alpha} = \Omega_m (1+z)^3$$

where $r_c = H_0^{-1} \Omega_{r_c}^{\frac{1}{\alpha-2}}$ and (6):

$$\implies \frac{H^2}{H_0^2} - (1-\Omega_m) \frac{H_\alpha}{H_0^\alpha} - \Omega_m(1+z)^3 = 0 \tag{8}$$

Figure 4. Distance luminosities determined by αDGP model over a range of redshifts, $z \in [0, 2]$, for various $\alpha$ and $\Omega_m = 0.3$.

# 4 Results

In the following section we are going to discuss the heart of this thesis: the parameter selection for ΛCDM and αDGP models, respectively. In the first model we are trying to pinpoint a value for the matter and dark energy densities and in the second model, which eliminates the need for a cosmological constant entirely, we want to find an optimal fit of the matter density and the parameter $\alpha$, which describes a sub-leading correction to the standard cosmology.

The data used in the following analysis, is provided the Supernova Cosmology Project (SCP). The sample contains redshift ($z_i$), magnitude ($m_i$), and magnitude error ($\sigma_{m_i}$) of 580 passable SNe Ia, including high-redshift $z \gt 1$ observations from the HST Cluster Survey (hereafter, “Union”) [13].

In order to determine whether the models fulfill our theoretical expectations, we use the $\chi^2$ method for goodness of fit. Furthermore, we aim to minimize the $\chi^2$ statistic by analytical and non-analytical means, when possible. In the second part of ΛCDM to lay grounds for the αDGP model, we employ a technique called marginalization in order to eliminate a parameter from the model.

According to Bayes’ Theorem, if we know the a priori probability of a parameter set, $p(\omega)$, we can calculate the probability distribution given the observations a posteriori.

$$p(\theta \big\vert D) = \frac{p(D\big\vert\theta) \cdot p(\theta)}{p(D)}$$

We introduce the likelihood function $[0, 1] \ni L_D(\theta) = p(D \big\vert \theta)$ that we set out to maximize in order to find the parameter set $\theta = \lbrace\Omega_m, \Omega_\Lambda, M, …\rbrace$ that best fits the model. The probability of the observations is uniformly distributed $\implies p(D) = 1$.

In the law of large numbers, we can assume that the magnitude data and its error follow a Gaussian distribution. Thus,

$$p_i(\theta \big\vert D) = \frac{1}{\sqrt{2\pi\sigma_i^2}} \exp\Bigg( -\frac{(m_i -m(z_i))^2}{2\sigma_i^2} \Bigg)$$

We start out unbiased about our parameters, i.e. we have no prior knowledge ($p(\theta) = 1$), and therefore the posterior probability equals the likelihood function.

$$L_D(\theta) \varpropto \exp\Bigg( - \displaystyle\sum_{i=1}^N \frac{(m_i - m(z_i))^2}{2\sigma_i^2} \Bigg) \equiv \exp\Bigg( - \frac{1}{2} \chi^2(\theta) \Bigg)$$

So, by minimizing $\chi^2(\theta) = -2 \ln L_D(\theta)$, we in fact maximize the $L_D(\theta)$ function and find best-fitted model parameters. With this fundamental understanding of best-fit statistics, we shall dive into the actual data analysis.

## 4.1 ΛCDM

For the flat ΛCDM model, we calculate the prediction based on the redshift data from Union and compare it to the actual observed magnitude. We replace $\Omega_\Lambda$ in the Friedmann equation (2) so that the entire model only depends on $\Omega_m$, $M$, and redshift $z$. From there we can calculate the magnitude and the $\chi^2$ as follows:

$$H^2(z) = H_0^2[\Omega_m(1+z)^3 + (1- \Omega_m)]$$

$$d_L(z) = (1+z) \int_0^z \frac{d\tilde{z}}{H(\tilde{z})}$$

$$m = M + 5 \log_{10}d_L +25$$

$$\chi^2 = \displaystyle\sum_{i=1}^N \frac{(m_i - m(z_i, \Omega_m, M))^2}{\sigma_{m_i}^2}$$

Figure 5. Observations of SNe Ia apparent magnitude (and observed errors) compared with the theoretical apparent magnitude with best-fitted parameters for $\Omega_m$ and absolute Magnitude $M$.

The minimum of $\chi^2$ is found through a non-analytical method – iterating over various values for $\Omega_m$ and $M$ – and then recording the respective $\chi^2$ results until the $\Omega_m - M$ sequence ends and a minimum is found. We plot the $1\sigma$ − $3\sigma$ 2D contours of the $\chi^2$ function and its sole input parameters $\Omega_m$, $M$.

Figure 6. $1\sigma$ – $3\sigma$ $\chi^2$ contours on a 2D plane with fitted parameters $\Omega_m$ and $M$.

The optimal allocation of non-relativistic matter and dark energy in the flat ΛCDM based on Union is

\begin{aligned} \Omega_m &= 0.32 \cr M &= 0.02 \cr \Omega_\Lambda &=0.68 \end{aligned}

Errors follow the Gaussian distribution and are in a narrow band around the minimum (see Figure 6).

In the first section, we have established the magnitude as $m = M + 5\log_{10}d_L + 25$; however, the absolute magnitude $M$, the factor 25, and the Hubble parameter (nested inside $d_L$) are mere constants that we can actually summarize in $\mathcal{M} = M - 5 log_{10} H_0 + 25$. We adjust the distance luminosity accordingly, where $H_0$ can easily be extracted from the Friedmann equation ($H(z) = H_0 \cdot E(z)$).

$$d_L(z) = \frac{(1+z)}{H_0} \int_0^z \frac{\tilde{z}}{E(\tilde{z})} \equiv \frac{D_L(z)}{H_0}$$

After plugging these new definitions in $\chi^2$ and performing a simple quadratic expansion, we are left with a $\chi^2$ function that has successfully extracted the constant $\mathcal{M}$,

$$\implies \chi^2 = f_1 -2 \mathcal{M} f_0 + \mathcal{M}^2 c_1$$

To follow the convention set forth in [14], we can calculate the new likelihood function by integrating over $\mathcal{M}$ and get

$$\int \mathcal{L}_D(\Omega_m, \mathcal{M})d\mathcal{M} \varpropto \mathcal{L}_D(\Omega_m) \varpropto \exp \Bigg( -\frac{1}{2} \Big( f_1 - \frac{f_0^2}{c_1} \Big) \Bigg)$$

$$\tilde\chi^2 = f_1 - \frac{f_0^2}{c_1}$$

\begin{aligned} c_1 &= \displaystyle\sum_{i=1}^N \frac{1}{\sigma_{m_i}^2} \cr f_0 &= \displaystyle\sum_{i=1}^N \frac{5 \log_{10} \mathcal{D}_L(z_i) - m_i}{\sigma_{m_i}^2} \cr f_1 &= \displaystyle\sum_{i=1}^N \frac{(5 \log_{10} \mathcal{D}_L(z_i) - m_i)^2}{\sigma_{m_i}^2} \end{aligned}

The data approves of this marginalization (denoted with a tilde) when compared to the “full” ΛCDM results:

$$\tilde\chi^2 = 570.2630 \qquad\qquad \chi^2 = 570.3912$$ $$\tilde\Omega_m = 0.33 \qquad\qquad \Omega_m = 0.32$$ $$\tilde\Omega_\Lambda = 0.67 \qquad\qquad \Omega_\Lambda = 0.68$$

## 4.2 αDGP

In the previous part we have not eliminated $\mathcal{M}$ selflessly, but with a motive to introduce a new parameter, namely $\alpha$. Now we can amend the model quite easily because from one of our special cases ($\alpha = 0$), we should get the ΛCDM cosmology. Therefore we can just write $\mathcal{D}_L = \mathcal{D}_L(z,\alpha) \implies \chi^2(z,\alpha)$.

Except for the special cases (see Figure 4 and Figure 7), the new modified extended Friedmann equation cannot be solved analytically. We have to rephrase (8) to $f(u)$ and find its roots (the values for parameter $u = \frac{H}{H_0}$ that make the function equal to zero):

$$f(u) \coloneqq u^2 - (1 - \Omega_m) u^\alpha - \Omega_m (1+z)^3 = 0 \tag{9}$$

Figure 7. Relative magnitude as determined by αDGP model for the best-fitted 𝛼 and two special models over the range of the observed redshifts from the SNe Ia data

With the secant method – a finite approximation of the Newton method – we can find the roots for all of Union, dynamically. Explicitly,

$$u_n = u_{n-1} - f(u_{n-1}) \frac{u_{n-1} - u_{n-2}}{f(u_{n-1}) - f(u_{n-2})}$$

where $n$ are the maximum number of iterations until a root has to be found. This is also called a recursive function as it uses the last calculated value as the base for its next iteration. The optimization function newton from the SciPy library is utilized to achieve this for the entire dataset [21].

Having found the zeroes of (9), we can proceed similar to the second half of ΛCDM, just that $\mathcal{D}_L$, and by association, $f_0$ and $f_1$ are now dependent of $\Omega_m$ and $\alpha$.

We find the minimum of $\chi^2$ by iterating over a range of $\Omega_m$ and $\alpha$ values.

$$\chi^2 = 562.2323$$ $$\Omega_m = 0.27$$ $$\alpha = 0.1$$

Figure 8. $1\sigma$ – $3\sigma$ $\chi^2$ contours on a 2D plane with fitted parameters $\Omega_m$ and $\alpha$.

Figure 9. Detail view of the $\alpha \in [0,2)$ region. The plot becomes very imprecise as $\alpha \to 2$ and $\Omega_m \to 0$, where no matter can exist in the universe.

The distribution of errors of αDGP is not Gaussian; although, the minimum of 𝛼 is extremely close to ΛCDM ($\alpha =0$), this analysis implies that based on Union within a confidence interval of 99.5% ($3\sigma$), 𝛼 can take up any value up to 0.75 ($\alpha \le 0.75$). The flat DGP model ($\alpha =1$) can be ruled out as possible theory for explaining the accelerated universe. Unfortunately though, we cannot invalidate either model from the $\chi_{min}^2$ results alone. On one hand, ΛCDM confirms our assumption about the composition of the flat universe with a roughly 1:2 matter-dark energy distribution. On the other hand, αDGP establishes itself as a solid contender explaining the late-time acceleration for our current universe with a similar cosmic matter density.

# 5 Conclusion

In this thesis we have compared a proven cosmological theory, ΛCDM, that dates back to Einstein and a newer alternative approach, αDGP, that modifies Einstein gravity at its core. In an attempt to maximize the likelihood function, i.e. maximizing the probability of the parameter set after taking the observations into account, we found that even though the contours for $\chi_{ΛCDM}^2$ and $\chi_{αDGP}^2$ look very different (_Figure 6_ and _Figure 8_, respectively), they have a minimum near the expected, theoretical value for the current non-relativistic matter density, $\Omega_m \sim 0.3$. One difficult task in the analysis of αDGP is that the errors do not follow a Gaussian distribution. If we know more about the distribution of $\alpha$ and the dependence on the growth factor $f(z)$ , especially the long tail in the negative region (see _Figure 8_), we could determine the cosmic growth history for αDGP that also explains the distance luminosities we observe, like [15] for ΛCDM. The popular alternative $f(R)$ gravity model can also be reproduced from αDGP in the negative-$\alpha$ space [18].

We cannot strip the title “standard cosmology model” from ΛCDM just yet. As an alternative approach to a self-accelerating expansion of the universe, αDGP does a phenomenal job in explaining and getting the cosmology right. For the sake of this thesis, we only considered supernova measurements from SCP; however, to differentiate the two models further as well as to create a more complete picture of the universe, one may want to look at other statistically significant cosmological data, such as cosmic microwave background anisotropy, baryon acoustic oscillations, and even gamma ray bursts.

In a stimulus among modified gravity theories, αDGP does not stand unchallenged; one may want to investigate $f(T)$ and $f(R)$ models [10], which directly modify intrinsic components – the torsion in the Lagrangian and the Ricci scalar itself, respectively – of the models’ Einstein-Hilbert action.

At last, we can always do more. Surely our models have room for improvement, but mainly we are limited by the precision of the telescopes available to us. The study of our cosmos is a long-term game – the universe may be expanding but it will not disappear. We have to persevere.

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