# Large Scale Cosmological Anomalies and Inhomogeneous Dark Energy

## Abstract

**:**

## 1. Introduction

**The Cosmic Microwave Background (CMB) Spectrum:**The angular power spectrum of CMB primordial perturbations [2] is in good agreement with the predictions of ΛCDM. However, a few issues related to the orientation and magnitude of low multipole moments (CMB anomalies) constitute remaining puzzles for the standard model [3,4,5,6,7,8,9,10,11].**Accelerating expansion:**Cosmological observations using standard candles [Type Ia Supernovae (SnIa)] [14] and standard rulers (Baryon Acoustic Oscillations) [15] to map the recent accelerating expansion rate of the universe are consistent with the existence of a cosmological constant. No need has appeared for more complicated models based on dynamical dark energy or modified gravity, despite of the continuously improved data. The likelihood of the cosmological constant vs. more complicated homogeneous models has been continuously increasing during the past decade [16].

**Power Asymmetry of CMB perturbation maps:**A hemispherical power asymmetry in the cosmic microwave background (CMB) on various different angular scales (multipole ranges) has been detected [22,23,24,25]. The power in all multipole ranges is consistently found to be significantly higher in the approximate direction towards Galactic longitude and latitude ($l={237}^{\circ},b=-{20}^{\circ}$) than in the opposite direction. A more recent study of the WMAP9 data has found a hemispherical directional dependence of ΛCDM cosmological parameters which is maximized at the direction ($l={227}^{\circ},b=-{27}^{\circ}$) at the multipole range 2–600 and is statistically significant at the $3.4$σ level [26]. A related asymmetry, is the Maximum Temperature Asymmetry (MTA) defined as the maximized temperature difference between opposite pixels in the sky which shows significant alignment with other apparently unrelated asymmetries [27].**Large Scale Velocity Flows:**Recent studies have indicated the existence of dipole velocity flows on scales of 100 h${}^{-1}$ Mpc [28,29] with magnitude about 400 km/s using a combination of peculiar velocity surveys, with direction towards $l\simeq {282}^{\circ}$, $b\simeq {6}^{\circ}$. Other studies [30] using the kinematic S-Z effect have found bulk flows on much larger scales (O(1 Gpc)) with magnitude 600–1000 km/s towards a similar direction. These results are inconsistent with the predictions of ΛCDM at a level of $99\%$. These studies however have been challenged by other authors which do not confirm these results but find peculiar velocities on these scales consistent with ΛCDM [31]. Even though these studies agree with the direction of the observed flow they disagree on the magnitude and errorbar of the measured velocities.**Alignment of low multipoles in the CMB angular power spectrum:**The normals to the octopole and quadrupole planes are aligned with the direction of the cosmological dipole at a level inconsistent with Gaussian random, statistically isotropic skies at 99.7% [4]. This inconsistency has been reduced by the recent Planck results to a level of about 98% [22] (the exact level varies slightly depending on the foreground filtering method).**Large scale alignment in the QSO optical polarization data:**Quasar polarization vectors are not randomly oriented over the sky with a probability often in excess of 99.9%. The alignment effect seems to be prominent along a particular axis in the direction $(l,b)=({267}^{\circ},{69}^{\circ})$ [32,33,34].**Anisotropy in Accelerating Expansion Rate:**Recent studies of the accelerating cosmic expansion rate using SnIa as standard candles have indicated that an anisotropic expansion rate fit by a dipole provides a better fit to the data than an isotropic expansion rate at the 2σ level [35] Interestingly, this dark energy dipole is abnormally aligned [35] with the fine structure constant dipole discussed below.**Spatial dependence of the value of the fine structure constant**α: A spatial cosmic variation of the fine structure constant has been recently identified on redshifts up to $z\simeq 4.2$ by analyzing the absorption spectra of quasars. This anisotropy analysis of the fine structure constant α [36,37] is based on a large sample of quasar absorption-line spectra (295 spectra) obtained using UVES (the Ultraviolet and Visual Echelle Spectrograph) on the VLT (Very Large Telescope) in Chile and also previous observations at the Keck Observatory in Hawaii. An apparent variation of α across the sky was found. It was shown to be well fit by an angular dipole model $\left(\right)open="("\; close=")">\frac{\Delta \alpha}{\alpha}=Acos\theta +B$ where θ is the angle with respect to a preferred axis and $A,B$ are the dipole magnitude and an isotropic monopole term. The dipole axis was found to point in the direction $(l,b)=({331}^{\circ},-{14}^{\circ})$ and the dipole amplitude A was found to be $A=(0.97\pm 0.21)\times {10}^{-5}$. The statistical significance over an isotropic model was found to be at the $4.1$σ level.**Large Quasar Group:**An elongated structure of quasars with long dimension about 1240 Mpc and mean redshift $\overline{z}=1.27$ has recently been discovered [38]. This structure is a factor of about three larger than the previously known largest structure (Sloan Great Wall ($\overline{z}=0.073$ and comoving size 450 Mpc [39]) and appears to be inconsistent with the cosmological principle in the context of the standard ΛCDM model in the sense that it is much larger than the scale of homogeneity in the context of ΛCDM (260–370 Mpc [40]).

**Figure 1.**Directions in galactic coordinates for the α (blue) and Dark Energy (green) dipoles, for the Dark Flow direction (red) and for the direction of Maximum Temperature Asymmetry (MTA) in the 7 years ILC CMB map degraded to ${N}_{side}=8$ (yellow). The opposite corresponding directions are also shown (from [27]).

- Dark matter perturbations on a few Gpc scale [51,52]. For example an off center observer in a 1 Gpc void would experience the existence of a preferred cosmological axis through the Lemaître-Tolman-Bondi metric [53,54,55,56,57]. Within this framework, an additional dark energy component is not needed to secure consistency with the cosmological data that indicate accelerating expansion. The basic idea is that the increased expansion rate occurs locally in space rather than at recent cosmological times, a fact that can be achieved by assuming a locally-reduced matter energy density [58,59]. Thus, the observer is placed close to the center of a giant void with dimensions of a few Gpc [60]. Even though this approach is free of dark energy, it is by no means free of fine tuning. Apart from the unnatural assumption of giant-size Gpc voids, which are very unlikely to be produced in any cosmology, these models require the observer to be placed within a very small volume at the center of the void (about ${10}^{-6}$ of the total volume of the void). A slightly off-center observer, however, will naturally experience a preferred cosmological direction (towards the center of the void), which may help to resolve some of the observational puzzles of ΛCDM discussed above.Such matter perturbations could be induced by statistically anisotropic or non-gaussian primordial perturbations [61,62,63,64]. For example, inflationary perturbations induced by vector fields [65,66,67,68]. Note however that inflationary models with vector fields usually suffer from instabilities due to the existence of ghosts [69,70,71].
- Dark energy perturbations on scales comparable to the horizon. Even though the sound speed for dark energy is close to unity implying that it can not cluster on scales much smaller than the horizon, it can still produce observable effects due to clustering on horizon scales [77,78]. In addition the possible recent formation of topological defects with Hubble scale core (topological quintessence) could also behave as inhomogeneous dark energy [79].

- There is no simple physical mechanism to generate such large voids [84].
- The fit for the expansion rate as a function of redshift is worse than ΛCDM requires significant fine tuning of the void profile to obtain a comparable fit with ΛCDM [86].
- These models predict significant peculiar velocities for distant galaxies within the void. Such velocities could have been detected in CMB maps through the kinematic S-Z effects. The fact that such large radial velocities are not observed imposes the most severe class of constraints in this class of models [82].

- It is a new generic generalization of ΛCDM including ΛCDM as a special case.
- It naturally violates the cosmological principle on large cosmic scales and predicts a preferred axis for off center observers. Thus it has the potential to address at least some of the above discussed cosmic anomalies.
- There is a well defined physical mechanism that can give rise to this type on dark energy inhomogeneities. This mechanism will be discussed in some detail in Section 3. It is based on applying the principles of topological inflation [87] to the case of late-time acceleration. According to the idea of topological inflation, the false vacuum energy of the core of a topological defect can give rise to accelerating expansion if the core size reaches the Hubble scale when gravity starts dominating the dynamics. Thus, for example, a recently formed global monopole with appropriate scale of symmetry breaking and coupling could naturally produce a Hubble-scale, spherically symmetric, isocurvature dark energy overdensity. By analogy with topological inflation, this mechanism may be called topological quintessence.

## 2. A Review of Cosmic Anomalies

#### 2.1. Power Asymmetry of CMB Perturbation Maps

- A large part (if not all) of the asymmetry appears to be due to the ISW effect which occurs at late times and is not related to the primordial nature of the CMB perturbations [94].
- The power spectrum of large scale structure does not show evidence for such an dipole asymmetry on smaller scales. This also hints towards a possible late time origin of the asymmetry.

#### 2.2. Alignment of Low Multipoles in the CMB Angular Power Spectrum

#### 2.3. Large Scale Velocity Flows

#### 2.4. Large Scale Alignment in the QSO Optical Polarization Data

#### 2.5. Anisotropy in Accelerating Expansion Rate (Dark Energy Dipole)

**Figure 2.**The distribution of the Union2 SnIa datapoints in galactic coordinates along with the Dark Energy dipole direction are shown. The datapoints are split in three different redshift bins shown with different shapes. The direction of the α-dipole is also shown with a star. The light blue blob represents the 1-σ error on the Dark Energy dipole direction (from [35]).

#### 2.6. Anisotropy in the Values of the Fine Structure Constant α (α Dipole)

#### 2.7. Large Quasar Groups

## 3. Anisotropic/Inhomogeneous Dark Energy Models

- Deviate from ΛCDM mainly on large cosmological scales.
- Reduce to ΛCDM for certain values of its parameters.

#### 3.1. Anisotropic Dark Energy Models

#### 3.2. Inhomogeneous Dark Energy Models

**Figure 3.**An off-center observer in a spherical inhomogeneity sees a preferred direction of higher accelerating expansion rate.

**Figure 4.**The 1σ and 2σ contours for the parameters ${\Omega}_{M,\text{in}}$ (matter underdensity inside the void) and ${r}_{0}$ (size of the void) (from [86]).

- They can naturally provide a preferred axis by utilizing a slight displacement of the observer from the center of the spherical void.
- The coincidence problem is not present since a void can naturally develop at late times along with other structures.

- In the context of standard cosmology there is no simple mechanism to create Gpc matter voids without conflicting other cosmological data (e.g., CMB fluctuations).
- There is no simple matter density profile that will provide an equal or better quality of fit compared to ΛCDM [86].
- This class of models generically predicts large peculiar velocities of free electrons and clusters with respect to the CMB. These velocities could have been detected via inverse Compton scattering of CMB photons onto moving free electrons (kinematic Synyaev-Zeldovich effect). The existing bounds on such velocities effectively rules out this class of models [82,83].

- The standard cosmological model ΛCDM assumes the existence of homogeneous matter on large scales, the validity of general relativity and the existence of homogeneous dark energy with constant in time energy density (cosmological constant). The generalization of most of these assumptions has been extensively studied in the literature. The main motivation for such generalizations originates at the coincidence problem expressed as a “Why now?” problem: “Why was the energy scale of the cosmological constant tuned to such low values so that it starts dominating at the present time?”. For example general relativity has been generalized to various models of modified gravity, dark energy was allowed to evolve in time as a scalar field (quintessence), matter was allowed to be inhomogeneous on Gpc scales (void models). The least studied generalization is the one based on allowing dark energy to become inhomogeneous on Gpc scales. Such a generalization could in principle address the coincidence problem if the later is expressed as a “Why here?” problem: “Why is the dark energy density in our horizon such that the universe has recently started its accelerating expansion?”. In the same manner that the answer to the “Why now?” question could be time dependence, the answer to the “Why here?” question could be spatial dependence of dark energy density. Therefore, the consideration of inhomogeneous dark energy constitutes a generic generalization of the standard ΛCDM models which reduces back to ΛCDM for a inhomogeneity scale that exceeds the current horizon scale. It is therefore interesting to use cosmological observations to impose constraints on the two basic parameters of this class of models: the scale of the inhomogeneity and the magnitude of the dark energy density inhomogeneity.
- As in the case of the dark matter void, an off-center observer in a spherically symmetric dark energy inhomogeneity will naturally experience an anisotropy in the accelerating expansion rate. In the case of inhomogeneous dark energy the scale of the inhomogeneity is allowed to be as large as the horizon scale (ΛCDM limit). For large enough inhomogeneity scale, the observer is allowed to be displaced significantly from the center without experiencing a large CMB dipole. Thus, inhomogeneous dark energy models are less subject to fine tuning with respect to the location of the observer [86].
- Due to negative pressure and speed of sound close to unity, dark energy inhomogeneities are not easy to sustain. However, there is a well defined physical mechanism (topological quintessence [79]) that can lead to sustainable large scale dark energy inhomogeneities, supported by topological considerations. This mechanism can be viewed as a generalization of topological inflation [87] in which early time accelerating expansion (inflation) takes place in the core of a topological defect due to the topologically trapped vacuum energy.

#### 3.3. Topological Quintessence

**Figure 5.**An off-center observer in the core of a global monopole will naturally experience an anisotropy in the expansion rate due to the higher vacuum energy density in the direction of the monopole center.

- Does the monopole energy density eventually dominate over matter in the monopole core?
- Does the possible monopole domination eventually lead to accelerating expansion in the monopole core?
- Can this cosmological expansion in the core fit the cosmological data?

**Figure 6.**Evolution of the dark energy (left panel) and properly normalized matter (right panel) density profiles.The profiles shown correspond to the initial time ${t}_{0}$ (dotted line), ${t}_{p}/3$ (dot-dashed line), $2{t}_{p}/3$ (dashed line), ${t}_{p}$ (solid line, present time). Notice the matter underdensity that develops in the monopole core while the monopole density appears to slowly collapse to the center in comoving coordinates. The radial unit is the monopole core size δ (from [79]).

**Figure 7.**Evolution of the scale factors $A(r,t)$ and $B(r,t)$ profile for $\eta =0.1$ (black lines) and $\eta =0.6$ (blue lines) from the time ${t}_{0}$ up to the present time ${t}_{p}$. A mesh of dots is superimposed to the curve corresponding to $\eta =0.1$, $t={t}_{p}$ to distinguish it from the curve obtained for $\eta =0.6$, $t={t}_{p}$. The profiles shown correspond to ${t}_{0}$ (dotted line), ${t}_{p}/3$ (dot-dashed line), $2{t}_{p}/3$ (dashed line), ${t}_{p}$ (solid line). Higher curves correspond to more recent times. The (rescaled) comoving region of accelerated expansion is slightly smaller in the case of $\eta <0.3$ since the monopole core does not expand in this case (from [79]).

**Figure 8.**Ratios ${H}_{A}(r,z)/{H}_{A}(r,0)$ and ${H}_{B}(r,z)/{H}_{B}(r,0)$ for $r=0$ (lower solid line), $r=0.5$ and $r=5$ (upper solid lines) along with the corresponding ${H}_{\Lambda}(r,z)/{H}_{0}$ for ${\Omega}_{\Lambda}=0.73$ (thick, dashed line) and for ${\Omega}_{\Lambda}=0$ (dot-dashed line). Also included for comparison ${H}_{\Lambda}(r,z)/{H}_{0}$ for the best fit values of ${\Omega}_{\Lambda}\left(r\right)$: ${\Omega}_{\Lambda}(0.5)\simeq 0.27$ (left panel) and ${\Omega}_{\Lambda}(0.5)\simeq 0.54$ (right panel). We have set $\eta =0.1$ (from [79]).

#### 3.4. Spherical Dark Energy Overdensity

**Figure 9.**The assumed matter and dark energy profiles are consistent with flatness and with matter domination outside the inhomogeneity. Such profiles would develop dynamically in the context of topological quintessence (see Figure 6).

**Figure 10.**The 1σ and 2σ parameter contours for the inhomogeneous dark energy (IDE) fluid model obtained using the Union 2 data, compared with the corresponding contours of the usual LTM matter void model. Notice that for an inhomogeneity size larger than the range of the dataset (about 4 Gpc), the model reduces to ΛCDM(${\Omega}_{m}=2.7$, ${\Omega}_{X}=0.73$. The transition scale was fixed to $\Delta r=$ 0.35 Gpc (from [86]).

**Figure 11.**The lightlike geodesics for an off-center depend on two additional parameters ${r}_{obs}$ and ξ.

**Figure 12.**The dependence of ${\chi}^{2}$ on the displacement ${r}_{obs}$ of the observer (left panel). Minimization with respect to the direction of the inhomogeneity center has been performed. We have considered four pairs (${r}_{0},{\Omega}_{X,\text{in}}$) that provide good fits for the on center observer (right panel). Notice that improved fit with respect to ΛCDM (horizontal blue dashed line) is obtained only for large scale inhomogeneities (${r}_{0}>3.5$ Gpc) (from [86]).

**Figure 13.**The predicted CMB dipole moment ${a}_{10}\left({r}_{obs}\right)$ for ${\Omega}_{X,\text{in}}=0.69$ and various values of the inhomogeneity scale ${r}_{0}$. The dashed line corresponds to the observed value (whose uncertainty is too small to be shown (from [86]).

**Figure 14.**The spatial fraction $f\left({r}_{0}\right)\equiv {\left(\frac{{r}_{obs-max}}{{r}_{0}}\right)}^{3}$ where the observer needs to be confined in order to be consistent with the value of the observed CMB dipole, for $({r}_{0},{\Omega}_{X,\text{in}})=(3.37\phantom{\rule{0.166667em}{0ex}}\mathrm{Gpc},0.69)$ (from [86]).

**Figure 15.**The predicted CMB maps of additional temperature fluctuations in galactic coordinates corresponding to the full map (

**a**); dipole (

**b**) quadrupole (

**c**) and octopole (

**d**). We have selected the direction of the preferred axis to coincide with the direction of the observed CMB dipole and used $({r}_{0},{\Omega}_{X,\text{in}})=(3.37\phantom{\rule{0.166667em}{0ex}}\mathrm{Gpc},0.69)$, ${r}_{obs}=30$ Mpc (from [86]).

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

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Perivolaropoulos, L.
Large Scale Cosmological Anomalies and Inhomogeneous Dark Energy. *Galaxies* **2014**, *2*, 22-61.
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Large Scale Cosmological Anomalies and Inhomogeneous Dark Energy. *Galaxies*. 2014; 2(1):22-61.
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