{'fraction_non_alphanumeric': 0.03630950079269943, 'fraction_numerical': 0.012412513050539422, 'mean_word_length': 4.317023026315789, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Anisotropic universe with anisotropic sources

We analyze the state space of a Bianchi-I universe with anisotropic sources. Here we consider an extended state space which includes null geodesics in this background. The evolution equations for all the state observables are derived. Dynamical systems approach is used to study the evolution of these equations. The asymptotic stable fixed points for all the evolution equations are found. We also check our analytic results with numerical analysis of these dynamical equations. The evolution of the state observables are studied both in cosmic time and using a dimensionless time variable. Then we repeat the same analysis with a more realistic scenario, adding the isotropic (dust like dark) matter and a cosmological constant (dark energy) to our anisotropic sources, to study their co-evolution. The universe now approaches a de~Sitter space asymptotically dominated by the cosmological constant. The cosmic microwave background anisotropy maps due to shear are also generated in this scenario, assuming that the universe contains anisotropic matter along with the usual (dark) matter and vacuum (dark) energy since decoupling. We find that they contribute dominantly to the CMB quadrupole. We also constrain the current level of anisotropy and also search for any cosmic preferred axis present in the data. We use the Union~2 Supernovae data to this extent. An anisotropy axis close to the mirror symmetry axis seen in the cosmic microwave background data from Planck probe is found.


Introduction
In standard cosmology it is assumed that the space-time is homogeneous and isotropic. After the discovery of temperature anisotropies of the Cosmic Microwave Background (CMB) and accelerating expansion of the universe, a standard cosmological model describing universe dominated by cold dark matter (CDM) and cosmological constant (Λ) is formulated, known as ΛCDM model or cosmic concordance model. Fluctuations in the temperature of CMB radiation are statistically isotropic in this model [1]. A central assumption in our standard model of cosmology (and in most cosmological models) is that the universe is homogeneous and isotropic upto small perturbations, and thus described by a perturbed Freidmann-Robertson-Walker (FRW) metric. Obersevations show that the temperature of the CMB is isotropic to a remarkable degree indicating that our universe is close to an Friedmann-Lemaitre (FL) model. According to Ehlers, Geren and Sachs (EGS) theorem [2], if the CMB temperature were exactly isotropic about every point in spacetime, then the universe would have to be exactly an FL model. This result is not directly applicable to CMB, because, CMB radiation is not exactly isotropic. The presence of temperature anisotropies in the CMB from the observations made by COBE/WMAP satellites are inconsistent with an exact homogeneous and isotropic FRW model [3].
The current CMB data supports an inflationary Big Bang model of cosmic origin for our universe. However at large angular scales, there are some anomalies observed in CMB data, such as, a low value for quadrupole power, alignment of quadrupole and octopole modes roughly in the direction of virgo cluster, ecliptic north-south power asymmetry, parity asymmetry between even and odd multipoles, almost zero correlations on large angular scales of CMB and an anomalous cold spot (∼ 10 • diameter) in the southern galactic hemisphere of CMB sky [4]. Several solutions to the low multipole CMB anomalies have been put forth such as foregrounds/systematics, anisotropic space-times and exotic topologies [5]. Some of these supposed deviations seen in CMB data were addressed by the WMAP and Planck science teams also [6,7]. These deviations may be an evidence that we live in a globally anisotropic universe.
Generally, in a global anisotropic universe, during inflation, shear decreases and eventually it goes over to an isotropic phase with negligible shear [8]. In order to produce any substantial amount of shear in recent times one needs to induce anisotropy in the spacetime. One way to induce anisotropy is to have anisotropic matter present at the last scattering surface. It was shown earlier that the power suppression in CMB quadrupole, without affecting higher multipoles, can be explained by assuming anisotropic matter (magnetic fields) [9]. Earlier works on anisotropic cosmological models with anisotropic stresses can be found, for example, in Ref. [10]. Anisotropic dark energy as a possible solution to the cosmic acceleration as well as the large scale CMB anomalies is studied, for example, in Ref. [11]. The anisotropic sources can be a uniform magnetic field, cosmic strings or domain walls. A nanogauss scale magnetic field could be present today which would have been produced during inflation due to a lorentz-violating term to the photon sector [12]. We call this as "Lorentz Violation generated magnetic field (LVMF)". Other possibilities such as a Maxwell type vector field coupled to a scalar field is studied, for example, in Ref. [13].
In this paper our approach is not to deal with metric approach directly, but rather to adhere to an approach similar to orthonormal frame formalism [14]. This paper is organised as follows. In section 2, we write down the Einstein's equations in cosmic time. Geodesic equations in cosmic time are studied in section 3. In section 4, we do a fixed point analysis of all the evolution equations in terms of a dimensionless time variable (τ ), to study the asymptotic evolution of the state observables. Then, in section 5, we show the temperature patterns for CMB due to various anisotropic sources considered in our study. A more realistic model including the ordinary dark matter and dark energy components in addition to the anisotropic sources already considered, is studied in section 6. In section 7, we obtain the constrains on the fractional energy densites due to various components in the realistic model of the previous section, the level of shear, and also determine a preferred axis if present. Finally, we conclude our work in section 8.

Bianchi I universe
The way we study dynamical systems is that we first write evolution equations in terms of few dimensionless state observables. It is a very invaluable tool to obtain qualitative information about the solutions of the state space of Bianchi universes (see Ref. [14] and references therein). Here we concentrate only on Bianchi I model. First we derive the Einstein equations in metric approach. Then we write in terms of the state observables analogous to the orthonormal frame formalism [15]. In this approach one writes the field equations as first order differential equations. An advantage of this orthonormal frame approach is that the derivation of geodesic equation is easy compared to the metric approach.
We start with a Bianchi I line element with a residual planar symmetry in the y − z plane as We choose the diagonal energy momentum tensor of the form T µ ν = (ρ, −p a , −p b , −p b ), where p a = w a ρ and p b = w b ρ. The four kinds of anisotropic matter we study here are given in Table 1. The magnetic field configuration considered here as anisotropic source is well known [10]. We, then consider a particular configuration of infinite domain walls (spanning the y − z plane) stacked along the x−axis. In this case, average energy momentum tensor will take the form we just described. Similarly, the Matter w a w b Magnetic Field -1 1 Cosmic String -1 0 Domain Walls 0 -1 LVMF 1 0 Table 1: Matter content for different anisotropic sources considered in this paper.
average energy momentum tensor due to a distribution of long, thin, straight cosmic strings along x−axis which takes an anisotropic form, also serves as anisotropic matter source [16]. Finally, a magnetic field generated due to a lorentz-violating term in the photon sector, which we call LVMF in short, is also studied as an anisotropic source [12]. The Einstein equations for the metric given in Eq. [1] are The equation of continuity is given by, Here onwards we choose to set 8πG = 1 for the rest of this paper. This can otherwise be seen as rescaling, all the quantities in Eq. [2] with 1/ √ 8πG. Now, we rewrite the above set of equations, Eq. [2] and [3], in terms of the average expansion H = (H a + 2H b )/3 and shear h = (H b − H a )/ √ 3. The Einstein's equations and the equation of continuity in cosmic time become and a constraint equation given by We now introduce dimensionless variables H ′ , h ′ , t ′ defined as where H 0 and h 0 are the current values of Hubble and shear parameters respectively. Then, the equations for the state observables in terms of these dimensionless variables can be written aṡ where an overdot represents differentiation with respect to t ′ and Ω 0 = ρ 0 /3H 2 0 . Now, we solve the above equation numerically for all the cases listed in Table 1. The evolution of the dynamic variables H ′ , h ′ and ρ ′ are plotted in Fig. [1], [2] and [3]. For all these anisotropic matter sources, it turns out that H ′ decreases with increasing time t ′ . Uniform magnetic field and LVMF matter have almost similar rate of expansion, where as walls and strings have a higher rate of expansion. Similarly from Fig. [2], we see that isotropy can be attained at late times for all cases of anisotropic matter and shear becomes negligible at late times. Here we find that isotropy sets in much later for walls compared to the other three sources. Except for walls, in all other cases isotropy is achieved almost at the sametime. We observe from Fig. [3] that, the energy density also decreases much faster for magnetic field, LVMF and strings compared to walls. From Fig. [2], it can be observed that evolution of shear depends on the initial conditions. Signature change of shear can be seen for certain kind of anisotropic matter. This signature change may have implications in early universe. From the left plot of Fig. [2], we see that this change occurs for walls, whereas right plot of Fig. [2] shows a similar signature change but for cosmic strings, depending on a positive/negative shear in the beginning. However late time behaviours of H ′ , h ′ , ρ ′ in cosmic time are independent of initial conditions.

Null Geodesics in cosmic time t ′
As we know the geodesic equations are second order differential equations in metric approach. If we reperesent the same equation in terms of H and h, then the geodesics can be written as first order differential equations which are given by [17] where ǫ = k 0 is the energy of photon, k 1 , k 2 and k 3 are the three components of photon momentum. The variables ǫ and k i (i=1,2,3) satisfies the constraint equation In terms of cosmic time t ′ the above equations becomė where an overdot represents differentiation with respect to t ′ . In order to solve the null geodesics, Eq. [10], one needs to take care that the solutions simultaneously satisfy the constraint equation, Eq. [9]. We solve the geodesic equations numerically for two particular cases. Case 1 : In this case only k 1 is taken to be nonzero, and k 2 and k 3 are zero. Initial values are taken as k 0 (0) = 20 and k 2 (0) = k 3 (0) = 0. The constraint equation, Eq. [9], gives k 1 = ±k 0 = ±ǫ.
Here we choose the positive value for k 1 . The evolution of k 0 is shown in the left plot of Fig. [4] for an initial values of k 0 (0) = 20, for the four anisotropic sources we considered. Here we find that, if the initial value of k 1 is positive then for the entire evolution it remains positive. It's also true for negative initial vaues of k 1 . Other intial values are same as Fig. [1], [2] and [3]. We observe that k 0 decreases slowly for magnetic field and faster for walls. This can be understood from  H ′ decreases slowly for walls and faster for a magnetic field. Case 2: In this case we take k 1 (0) = k 3 (0) = 25.3 and, k 2 = k 3 following the residual planar symmetry of our metric. Other intial values are same as Case 1 i.e., k 0 (0) = 20. The plots for k 1 and k 3 are shown in Fig. [5]. In this case, we also observe that k 0 decreases slowly for magnetic field and faster for walls, where as k 1 decreases faster in case of strings and slowly in case of walls.

Fixed point analysis of the evolution equations
In the previous section, we studied the evolution of state observables in terms of a time variable anolagous to cosmic time. In order to study the asymptotic evolution of our Bianchi I universe with various anisotropic sources considered here, we resort to dynamical systems approach. So, we cast all the evolution equations in terms of dimesionless variables which give us a set of dynamical equations. The advantage of working in this approach is that we can show the cosmological dyanamics insensitive to the intial conditions. Here we define the dimensionless variables τ, σ as dτ /dt = H and σ = h/ √ 3H. These can be called as expansion normalised time and shear variables. Thus our Einstein's equations and the continuity equation are given by and where q = 2σ 2 + (1 + 3w)/2Ω, w = (w a + 2w b )/3 and Ω = ρ/3H 2 . Next, we write the null geodesic equations in terms of dimensionless variables K i = k i /k 0 (i=1,2,3). They are given by 3) satisfy the constraint equation Here we will analyze the fixed points of the full set of equations combining Einstein equations, equation of continuity and the geodesic equations [17]. Fixed points for σ can be easily obtained from Eq. [11] which are +1, −1 and 2(w b − w)/(1 − w). All the stable fixed points are given in Table [ 2]. Same conclusion can be drawn from solving these equations numerically. The evolution of σ and, K 1 and K 3 are plotted in Fig. [6] and [7]. Asymptotically, we can see that σ, and K 1 and K 3 evolve towards their corresponding stable fixed points. This whole analysis is independent of the initial conditions.

Kasner solutions
Exact solutions of Kasner type can be obtained for the Einstein equations corresponding to the fixed points of the shear parameter σ [18]. In order to find Kasner type solution at stable fixed points we Here an initial value of σ = 0.01 is taken.
and average scale factor H satisfies After substituting the power law form for the scale factors into Eq. [15] and Eq. [16] and solving, we get The Kasner solutions are summarised in Table 3. From the exact solutions we observe that in the case of a magnetic field, universe expands along only one dimension, while it's static in the other two dimensions. For cosmic strings, it accelerates in one direction and decelerates in the other two directions. Then, in the case of domain walls, it decelerates in one direction and accelerates in the other two directions. For LVMF, it contracts in one direction and decelerates in the other two directions.

Temperature Patterns
Assuming that the universe is a Bianchi type I with anisotropic matter, we find the contribution to the CMB temperature anisotropies due to these sources. It has been shown earlier that the total quadrupole anisotropy in the presence of a uniform magnetic field, can be small compared to that obtained from the standard ΛCDM model [9]. The tempearture of the cosmic microwave background as a function of the angular coordinatesn = (θ, φ) on the celestial sphere is given by [19], where T LSS is the temperature of the CMB at the surface of last scattering, which is assumed to be isotropic, and The direction cosines along a null geodesic K α in terms of the spherical polar angles (θ, φ) are given by K α (τ 0 ) = (cos θ, sin θ cos φ, sin θ sin φ) .
Substituting Eq. [20] and Eq. [21] into Eq. [19] we get The geodesic equation for K 1 is where τ 0 (= 0) and τ LSS (= −7) are the values of τ -today and at the time of decoupling -and It turns out that the quadrupole contribution is dominant for all the four cases of anisotopic matter considered here. The CMB quadrupole temperature patterns are given in Fig. [8]. We find that all other higher multipoles receive negligible contribution due to these sources. Hence we only show temperature patterns for l = 2 here. We also observe from Fig. [8] that the temperature maps for magnetic field and cosmic strings have similar patterns, whereas temperature maps for domain walls and LVMF show similar patterns.

A more realistic scenario
In this section we extend our analysis of the previous section by adding the usual dark matter and cosmological constant as dark energy, to our anisotropic matter in order to make it a realistic scenario. We do our analyis in τ −time which is useful to find stable fixed points. However, the form of the geodesic equation will be unchanged.
The evolution equations in τ −time for this model which includes anisotropic matter, (isotropic) dark matter and dark energy are given by where, , The evolution of σ, Ω IM , Ω Λ , Ω AM are given in Fig. [9] for the two cases Ω IM > Ω AM and Ω IM < Ω AM . The late time behaviour in all these cases is almost same except for domain walls. However small may be the value of Ω DW to start with (at last scattering surface), it turns out that Ω DW tends to rise at intermediate times, and can even dominates over isotropic matter at present time for some initial values. This may not be a viable scenario as we know that the current dominant matter content is of isotropic type. However the other anisotropic sources can give rise to a viable scenario. For the actual evolution of an initial ordinary (isotropic) dark matter dominated era with negligible anisotropic matter and shear, to the current era of dark enegy domination, the deceleration parameter is found to evolve from 1/2 to -1 at late times, as expected.

SN Ia constraints on cosmic shear and anisotropy
In this section, we will test our anisotropic Bianchi-I model with anisotropic sources using supernova data. We constrain the level of anisotropic matter density, shear, and also determine the cosmic preferred axis, if present, in addition to the current Hubble parameter, dark matter and dark energy (modelled as cosmological constant) density fractions, using the Type Ia supernovae (SN1a) objects in the Union 2 compilation [20] 1 . For this section, we use the evolution equations as determined from the line element which correspond to the frame where the anisotropy axis is along the z−direction. Thus, the mean scale factor and the eccentricity are given by The redshift (z) and luminosoty distance (d L ) of an SN1a object observed in the directionn = (θ, φ) are given by [11] 1 + z(n) and where α is the angle between the cosmic preferred axis (λ) and the supernova position (n), and 'c' is the speed of light. Hence, cos α =λ ·n. In order to obtain constrains on our Bianchi-I model with anisotropic matter and shear in addition to ΛCDM parameters, we fit the measured distance modulus of various SN1a objects that are provided in the Union 2 compilation to the theoretical distance modulus, by minimizing the χ 2 defined as where µ d i is the measured distance modulus of an SN1a object from data, µ th i is the theoretical distance modulus function involving various cosmological parameters, δµ d i is the measured uncertainty in the distance modulus of an SN1a object provided in the data, and the summation is over all SN1a objects given in the data (total 557 supernovae). τ -7 -6.5 -6 -5.5 -5 τ τ 7 -6.5 -6 -5.5 -5 τ τ Ω IM > Ω AM Ω IM < Ω AM Ω LVMF Figure 9: Ω vs τ : Late time behaviour of shear and density fractions of ΛCDM components in addition to the different anisotropic sources studied in this work, considered seperately.
The theoretical distance modulus is defined as where pc in the denominator stands for 'parsec', and d L is the luminosity distance defined in Eq. [30]. By using the parametrization H = 100 h Km.sec −1 /M pc = k h, where k contains all the dimensionfull constants and h being the dimensionless Hubble parameter, Eq.
[32] can be further simplified to give where µ 0 = 42.384, andd L , containing dimensionless quantities, is given bỹ In this section, we use 'h' to denote the dimensionless Hubble parameter. The luminosity distance,d L , (and thus the distance modulus µ) depends on the cosmological parameters The minimization is done in conjugation with solving the evolution equations wherew = (2 w a + w b )/3, δ w = w a − w b , σ is the shear and Ω AM , Ω IM , Ω Λ are the fractional energy densities due to anisotropic matter (AM ), ordinary isotropic dark matter (IM ) and the dark energy (Λ). The ′ denotes a derivative with respect to the mean scale factor 'A'. These equations are evolved from z = [0, z SN i ] for each supernova i with the initial conditions at z = 0 as A = 1, e = 0, and initial guess values for the parameters {h 0 , σ 0 , Ω AM 0 , Ω IM 0 , Ω Λ 0 ,λ}. The deceleration parameter is given by q 0 = h ′ /h| z=0 . These anisotropic matter source models in a Bianchi-I universe are compared with the standard concordance model using the luminosity distance relation given bỹ The results from supernova distance modulus fits for the model with various anisotropic sources in addition to ΛCDM, and for the model with standard ΛCDM components solely, are shown in Table [4]. The corresponding χ 2 as a function of various cosmological parameters in our anisotropic model, and only the ΛCDM paramters are shown in Fig. [10] and Fig. [11], respectively. In those figures, note that the χ 2 is shown only for the effective six parameters {h 0 , σ 0 , Ω AM 0 , Ω Λ 0 ,λ = (R A , D A )} of our  Figure 10: χ 2 as a function of the effective six parameters {h 0 , σ 0 , Ω AM 0 , Ω Λ 0 ,λ = (R A , D A )} for the anisotropic universe considered in this work. Ω IM 0 dependence is not shown, treating it as a dependent parameter that can be estimated from the constraint equation Ω IM = 1−σ 2 −Ω AM −Ω Λ . The reference co-ordinate system for the anisotropy axis is the Equatorial co-ordinate system. anisotropic model with anisotropic sources, standard model components with a Bianchi I background considered in this work, and the two ΛCDM parameters {h 0 , Ω Λ 0 } corresponding to the flat FRW universe with the usual dark matter and dark energy, respectively. The ordinary isotropic matter (Ω IM ), is treated as dependent quantity that can be estimated from the corresponding constrain equations.
We find an improvement of ∆χ 2 = −2.83 for the Cosmic Strings, and ∆χ 2 = −2.79 for Domain Walls, Magnetic fields and Lorentz-violating Magnetic Fields as anisotropic sources in our anisotropic model, compared to the standard ΛCDM sources in a flat FRW universe. As is evident from Fig. [10], there is a weak dependence of χ 2 on the anisotropy axis parametersλ = (R A , D A ). But we find the same preferred axis for the various equation of state parameterizations (for different anisotropic sources), which might be indicative of a hidden preferred axis becoming explicit, independent of the specific anisotropic parameterization used. Both the weak dependence of anisotropy axis and the overall χ 2 improvement may be remedied by a future SN1a compilation containing more number of supernovae. Interestingly the anisotropy axes we found here, is also approximately close to other preferred directions found in diverse cosmological data [21], as shown in Fig.[12]. The CMB mirror parity symmetry axis [7] is the closest one to the axes we found here, at a mere angular separation of 10 • .  Figure 11: Same as Fig. [10], but for standard ΛCDM model parameters. Figure 12: Various anisotropic axes found in different cosmological observations. The reference coordiante system is the Galactic co-ordinate system (l,b). The additional axes used here are tabulated in Table [ [12] in addition to the anisotropy axes found in our anisotropic model (see Table [ 4]) are listed here [7], [21]. The axis are given in Galactic co-ordinate system.

Conclusions
Here we studied the evolution of Bianchi-I universe in the presence of different types of anisotropic matter sources. First we analyzed the evolution of the state space variables H ′ , h ′ , ρ ′ , k 0 , k 1 , k 2 = k 3 in cosmic time. Depending on the initial conditions we found that there can be a sign change in the shear. This may have some interesting implications in early universe cosmology. Then we examined the evolution of dimensionless variables viz., σ, K 1 and K 3 in dimensionless time τ . In the τ −frame, we determined the fixed points of the full set of evolution equations including the geodesic equations. We also checked the stability of the fixed points numerically. In terms of τ variable, we get an analytic expression for the temperature anisotropies of the CMBR. Then we generated the temperature patterns of the CMB for the four different types of anisotropic matter considered here. We found that the contribution to the CMB temperature signal is mainly through the quadrupole. We also pointed out the difference between the CMB tempearture maps due to all cases of anoisotropic matter considered in this work. We then investigated a more realistic scenario where we included ordinary isotropic dark matter and cosmological constant as dark energy. We found that the universe asymptotically evolves to a deSitter universe.
We also constrained the parameters of this model using Supernovae Union 2 data. We also found a preferred axis from the data for all the four cases of anisotropic matter. This anisotropy axis is found to be same independent of the anisotropic source we used. It turns out that this axis is very close to the mirror symmetry axis found in CMB data [7].
It will be interesting to generalise our results to other Bianchi classes -A and B. In case of other Bianchi models the state space would be larger and we need to find different subspace of the state space and their stable fixed points. It will also be interesting to study the CMB polarisation anisotropy in these models.