Fluctuations in the Ginzburg–Landau Theory of Dark Energy: Internal (In)consistencies in the Planck Data Set

Recently we have explored the framework of a phase transiting dark energy (Banihashemi et al. 2019, 2020, 2021). This idea states that dark energy as a self interacting many-body system is sensitive to the cooling down of the universe and consequently undergoes a phase transition. To investigate this proposal, we have built a model, based on the literature of critical phenomena and more specifically the Ginzburg–Landau theory (Ginzburg & Landau 1950). Therefore we have dubbed our model Ginzburg–Landau Theory of Dark Energy or just GLT for the sake of brevity. So far, we have mostly focused on GLT at the background level and showed that to some extent, we can address the famous H₀ tension ¹ via the dynamics that this model implies on dark energy. However, producing some specific sort of dynamical dark energy is the least thing this model can predict. In fact, going beyond mean-field approximation and considering the spatial variations of dark energy, opens a rich area of phenomenology and at the same time makes the model more falsifiable. This paper is our first attempt to investigate GLT beyond the mean-field regime. We are particularly interested in the spatial features of our model since there are several spatial oddities in cosmological observations. For instance, it has been reported that there is an inconsistency between low and high multipoles of the CMB temperature angular power spectrum (Planck Collaboration et al. 2020a). According to what the Planck team has revealed, when one puts constraints on flat ΛCDM parameters, using ℓ ≤ 801 of the CMB temperature angular power spectrum, the resulted confidence regions are inconsistent with the case when they are constrained using the ℓ ≥ 802 range. Also, they have shown that by allowing either Ω_k or the lensing amplitude, A_L , to vary as a free parameter, this discrepancy goes away. But none of these extensions to ΛCDM are really a solution for this issue. First, A_L ≠ 1 is not favored theoretically. It only gives a phenomenological hint about the cause of the problem. In fact, it introduces the so-called lensing anomaly, which is related to low- and high-ℓ inconsistency. Second, as again has been asserted by the Planck team, CMB data itself prefer a closed universe with nonzero Ω_k (Planck Collaboration et al. 2020a). But it has been shown that a nonzerospatial curvature can raise a crisis in cosmology (Di Valentino et al. 2020; Handley 2021), as the remaining cosmological observables are in strong disagreement with this curvature. There are several promising ideas for these problems to be alleviated. For example, in Moshafi et al. (2021), authors have shown that the $\ddot{u}{\rm{\Lambda }}$ CDM model relaxes the low- and high-ℓ inconsistency and the CMB lensing tension simultaneously, even better than the ΛCDM+A_L model. In Farhang & Khosravi (2021) and Khosravi & Farhang (2022), it has been shown that a phenomenological gravitational phase transition can address these concerns too. In Esteban et al. (2022), it has been shown that modifying the neutrino equation of state due to some unknown long-range interaction mimics A_L > 1 and hence alleviates the lensing anomaly.

In this work, we are going to investigate the time and scale dependence of dark energy patches in the GLT framework to see if they can relax the lensing anomaly and hence low- and high-ℓ inconsistency in the Planck data set. In Section 2 this model is reviewed and in Section 3, the contribution of dark energy patches in the lensing potential power spectrum is studied. In Section 4 we explain the data sets we have used and also the results. Finally, we end this paper with a few concluding remarks.

In this model we assume that a scalar field, say ϕ, is responsible for the dark energy evolution, in a way that

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{\Lambda }}}(z)=\langle \phi ({\boldsymbol{r}},z)\rangle ,\end{eqnarray} \tag{ 1 }$$

where the average is over space; the so-called Landau free energy governing this field is

$$\begin{eqnarray}&&L=\int {d}^{3}{\boldsymbol{r}}\left[\displaystyle \frac{\gamma }{2}{\left({\rm{\nabla }}\phi \right)}^{2}+m\,t\,{\phi }^{2}+\displaystyle \frac{1}{2}\lambda \,{\phi }^{4}\right].\end{eqnarray} \tag{ 2 }$$

In the above expression t is the reduced temperature, $t\equiv \tfrac{T-{T}_{c}}{{T}_{c}}$, where T, the temperature of dark energy, is assumed to be proportional to inverse of the scale factor, i.e., T ∝ (1 + z), and T_c is a critical temperature at which transition happens and 〈ϕ( r , z)〉 takes a nonzero value; γ, m, and λ are some constants that we try to put constraints on. As is evident in Equation (2), we have supposed ${{\mathbb{Z}}}_{2}$ symmetry for the Landau free energy; i.e., there is not any odd power of ϕ. This implies for the phase transition to be of second order and continuous; therefore the density changes smoothly in time. Also if we want to have an interaction term, we should have added a term like H( r , z)ϕ( r , z), where H( r , z) represents all the external fields that might have an interaction with ϕ( r , z). The behavior of the field is governed by the equation of motion deduced by demanding L to be stationary with respect to variations of ϕ:

$$\begin{eqnarray}&&\displaystyle \frac{\delta L}{\delta \phi }=0\Longrightarrow -\gamma {{\rm{\nabla }}}^{2}\phi +2{mt}\phi +2\lambda {\phi }^{3}=0.\end{eqnarray} \tag{ 3 }$$

If we consider ϕ within the mean-field approach, ϕ( r , z) = 〈ϕ( r , z)〉 ≡ M(z), then it should obey the following equation:

$$\begin{eqnarray}&&2mt\,M+2\lambda \,{M}^{3}=0,\end{eqnarray} \tag{ 4 }$$

which implies

$$\begin{eqnarray}&&M(t)=0\quad \mathrm{or}\quad M(t)=\pm \sqrt{-\displaystyle \frac{{mt}}{\lambda }}.\end{eqnarray} \tag{ 5 }$$

The case M = 0 corresponds to T > T_c (or t > 0). This means at z > z_t , the density of dark energy is effectively zero. The two other cases belong to the temperatures below T_c (or t < 0); after spontaneous symmetry breaking, M takes one of these two possible values. We suppose M takes the positive one ² and also, the assumption of spatial flatness of the universe fixes the ratio m/λ and we have

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{\Lambda }}}(z)={{\rm{\Omega }}}_{{\rm{\Lambda }}}\sqrt{\displaystyle \frac{{z}_{t}-z}{{z}_{t}}},\end{eqnarray} \tag{ 6 }$$

where Ω_Λ = 1 − Ω_m − Ω_r and z_t are the fractional density of dark energy today and the critical redshift corresponding to the transition temperature, respectively. So our model, at the background level, has one extra free parameter more than flat ΛCDM: z_t ; the Friedman equation takes its new form:

$$\begin{eqnarray}&&{H}^{2}={H}_{0}^{2}\left[{{\rm{\Omega }}}_{m}{\left(1+z\right)}^{3}+{{\rm{\Omega }}}_{r}{\left(1+z\right)}^{4}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}(z)\right].\end{eqnarray} \tag{ 7 }$$

This is almost all about GLT at the background level (Banihashemi et al. 2021) and one can easily put constraints on the free parameters using background data that are mostly geometrical. But we are also interested in the predictions of this model when the "fluctuations" are also considered. In principle, ϕ( r , z) can have spatial variations or "patches of dark energy with different densities." ³ This patchy pattern or fluctuations in the order parameter is very similar to different magnetic areas in a ferromagnetic substance: each region has its own magnetic orientation and one can partition the substance into equimagnetic areas. In addition to these fluctuations, there are also tiny "perturbations" on top of the average of the order parameter in every patch. In this work, we will not go into perturbations in detail and only stick to fluctuations, as they have the dominant effect. These fluctuations, whose statistics are given by the Ginzburg–Landau theory, affect cosmological observables. Among them, the change in the CMB lensing pattern intrigued us the most and is studied here. Since these patches have different densities, they induce extra gravitational potential in the perturbed FLRW metric and the trajectories of photons are changed, accordingly. In the following sections, we try to investigate this effect. Before doing so, we will derive the power spectrum of the field ϕ, which corresponds to the power spectrum of the dark energy density fluctuations. Basically, we are interested in the two-point correlation function of the field ϕ. Having the partition function of the system as (Goldenfeld 1992)

$$\begin{eqnarray}&&{ \mathcal Z }=\int { \mathcal D }\phi \,{e}^{-\beta L[\phi ]},\end{eqnarray} \tag{ 8 }$$

where again, L is the so-called Landau free energy, but this time with the external interaction term H( r )ϕ( r ),

$$\begin{eqnarray}&&\begin{array}{l}L=\int {d}^{3}{\boldsymbol{r}}\left[\displaystyle \frac{\gamma }{2}{\left({\rm{\nabla }}\phi ({\boldsymbol{r}})\right)}^{2}\right.\\ \left.+{mt}{\phi }^{2}({\boldsymbol{r}})+\displaystyle \frac{1}{2}\lambda {\phi }^{4}({\boldsymbol{r}})-H({\boldsymbol{r}})\phi ({\boldsymbol{r}})\right].\end{array}\end{eqnarray} \tag{ 9 }$$

It easily follows that the average of the field or its one-point function is obtained as

$$\begin{eqnarray}&&\langle \phi ({\boldsymbol{r}})\rangle =\displaystyle \frac{1}{\beta }\,\displaystyle \frac{\delta \mathrm{ln}{ \mathcal Z }}{\delta H({\boldsymbol{r}})},\end{eqnarray} \tag{ 10 }$$

and the two-point function can be extracted as

$$\begin{eqnarray}\begin{array}{rcl}G({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} ) & \equiv & \langle \phi ({\boldsymbol{r}})\phi ({\boldsymbol{r}}^{\prime} )\rangle -\langle \phi ({\boldsymbol{r}})\rangle \langle \phi ({\boldsymbol{r}}^{\prime} )\rangle \\ & = & \displaystyle \frac{1}{{\beta }^{2}}\,\displaystyle \frac{{\delta }^{2}\mathrm{ln}{ \mathcal Z }}{\delta H({\boldsymbol{r}}^{\prime} )\delta H({\boldsymbol{r}})}.\end{array}\end{eqnarray} \tag{ 11 }$$

In the above expressions, β = 1/T and by δ we mean the functional derivative. In order to obtain an equation for $G({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )$, we can make use of the equation of motion for ϕ( r ) and vary it with respect to $H({\boldsymbol{r}}^{\prime} )$:

$$\begin{eqnarray}&&-\gamma {{\rm{\nabla }}}^{2}\phi ({\boldsymbol{r}})+2{mt}\phi ({\boldsymbol{r}})+2\lambda {\phi }^{3}({\boldsymbol{r}})-H({\boldsymbol{r}})=0\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\begin{array}{l}\mathop{\longrightarrow }\limits^{\delta /\delta H({\boldsymbol{r}}^{\prime} )}\\ \,\,\beta \,[-\gamma {{\rm{\nabla }}}^{2}+2{mt}+6\lambda {\phi }^{2}({\boldsymbol{r}})]\,G({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )=\delta ({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} ).\end{array}\end{eqnarray} \tag{ 13 }$$

This equation governs the two-point correlation function of ϕ. By going to Fourier space, it not only becomes easier to solve it, but also we directly arrive at the power spectrum that was desired. Depending on being above T_c or below it, ϕ follows one of the behaviors mentioned in Equation (5) and for the power spectrum we have

$$\begin{eqnarray}&&{{ \mathcal P }}_{\phi }(k,z)=\displaystyle \frac{1+z}{\gamma }\displaystyle \frac{1}{{k}^{2}+{\xi }^{-2}(z)},\end{eqnarray} \tag{ 14 }$$

where the correlation length, ξ(z), in any of the above or below T_c regimes, is defined as

$$\begin{eqnarray}\begin{array}{rcl}\xi (z) & \equiv & {\left(\displaystyle \frac{\gamma (1+{z}_{t})}{2\,m(z-{z}_{t})}\right)}^{1/2}\quad \mathrm{for}\quad z\gt {z}_{t},\\ \xi (z) & \equiv & {\left(\displaystyle \frac{\gamma (1+{z}_{t})}{4\,m({z}_{t}-z)}\right)}^{1/2}\quad \mathrm{for}\quad z\lt {z}_{t}.\end{array}\end{eqnarray} \tag{ 15 }$$

In order to compute the lensing potential power spectrum, first we need power spectra of Bardeen potentials. Bardeen potentials are related to the density fluctuations through the Poisson equation, which in Fourier space reads as follows:

$$\begin{eqnarray}\begin{array}{rcl}{k}^{2}{\rm{\Psi }} & = & 4\pi G{\left(1+z\right)}^{-2}(\delta {\rho }_{m}+\delta {\rho }_{{de}})\\ & = & 4\pi G{\left(1+z\right)}^{-2}\left[{\bar{\rho }}_{m}{\delta }_{m}+\displaystyle \frac{3{H}_{0}^{2}}{8\pi G}\delta \phi \right]\\ & = & 4\pi G{\left(1+z\right)}^{-2}\displaystyle \frac{3{H}_{0}^{2}}{8\pi G}\left[{{\rm{\Omega }}}_{m}{\left(1+z\right)}^{3}{\delta }_{m}+\delta \phi \right]\\ & = & \displaystyle \frac{3}{2}{H}_{0}^{2}\left[{{\rm{\Omega }}}_{m}(1+z){\delta }_{m}+\displaystyle \frac{\delta \phi }{{\left(1+z\right)}^{2}}\right].\end{array}\end{eqnarray} \tag{ 16 }$$

So for the two-point correlation functions we have:

$$\begin{eqnarray}&&\langle {\rm{\Psi }}{\rm{\Psi }}\rangle =\displaystyle \frac{9\ {H}_{0}^{4}}{4\ {k}^{4}}\left[{{\rm{\Omega }}}_{m}^{2}{\left(1+z\right)}^{2}\langle {\delta }_{m}{\delta }_{m}\rangle +\displaystyle \frac{\langle \delta \phi \delta \phi \rangle }{{\left(1+z\right)}^{4}}\right],\end{eqnarray} \tag{ 17 }$$

therefore

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal P }}_{{\rm{\Psi }}}(k;z,z^{\prime} ) & = & \displaystyle \frac{9\ {H}_{0}^{4}}{4\ {k}^{4}}\left[\Space{0ex}{3.73ex}{0ex}{{\rm{\Omega }}}_{m}^{2}{{ \mathcal P }}_{{\delta }_{m}}(k,z=\infty )(1+z){T}_{{\delta }_{m}}(k,z)\right.\\ & & \times \left.(1+z^{\prime} ){T}_{{\delta }_{m}}(k,z^{\prime} )+\displaystyle \frac{\sqrt{{{ \mathcal P }}_{\phi }(k,z)}}{{\left(1+z\right)}^{2}}\displaystyle \frac{\sqrt{{{ \mathcal P }}_{\phi }(k,z^{\prime} )}}{{\left(1+z^{\prime} \right)}^{2}}\right],\end{array}\end{eqnarray} \tag{ 18 }$$

where ${ \mathcal P }$ and ${T}_{{\delta }_{m}}$ are the power spectrum and matter transfer function, respectively. In the above calculation, we assumed that matter and dark energy fluctuation fields are independent and ignore the cross term. The power spectrum of dark energy fluctuations is as described in Equations (14) and (15). Therefore, the lensing potential power spectrum reads

$$\begin{eqnarray}&&{C}_{{\ell }}^{{\psi }_{{tot}}}={C}_{{\ell }}^{{\psi }_{m}}+{C}_{{\ell }}^{{\psi }_{{de}}},\end{eqnarray} \tag{ 19 }$$

where

$$\begin{eqnarray*}\begin{array}{rcl}{C}_{{\ell }}^{{\psi }_{m}} & \equiv & 36\,\pi \,{H}_{0}^{4}\,{{\rm{\Omega }}}_{m}^{2}\,\displaystyle \int \displaystyle \frac{{dk}}{{k}^{5}}\,{{ \mathcal P }}_{{\delta }_{m}}(k,z=\infty )\\ & & \times {\left[{\displaystyle \int }_{0}^{{z}_{d}}\displaystyle \frac{{dz}}{H(z)}(1+z){T}_{{\delta }_{m}}(k,z){j}_{{\ell }}(k\chi )\left(\displaystyle \frac{\chi -{\chi }_{* }}{\chi {\chi }_{* }}\right)\right]}^{2},\end{array}\end{eqnarray*}$$

and

$$\begin{eqnarray*}\begin{array}{rcl}{C}_{{\ell }}^{{\psi }_{{de}}} & \equiv & 36\,\pi \,{H}_{0}^{4}\,\displaystyle \int \displaystyle \frac{{dk}}{{k}^{5}}\\ & & \times {\left[{\displaystyle \int }_{0}^{{z}_{d}}\displaystyle \frac{{dz}}{H(z){\left(1+z\right)}^{2}}\sqrt{{{ \mathcal P }}_{\phi }(k,z)}\ {j}_{{\ell }}(k\chi )\left(\displaystyle \frac{\chi -{\chi }_{* }}{\chi {\chi }_{* }}\right)\right]}^{2}.\end{array}\end{eqnarray*}$$

So our model predicts an extra term for the lensing potential power, namely ${C}_{l}^{{\psi }_{{de}}}$. In the following section we try to see if this extra term has the features desired by the data or not. For this purpose, we implemented our model into the publicly available code, CAMB (Lewis et al. 2000), to calculate the two-point statistics of the cosmic background radiation anisotropies. We did so by modifying both the expansion history of the universe and the source of the Poisson equation, by adding the fluctuation amplitude of dark energy in each scale and time, k, z, as a new source for the gravitational potential. Since there is not any nongravitational interaction between dark energy and other fluids, we left the Boltzmann equations as they are in ΛCDM model. To sample the parameter space and find the confidence regions of the free parameters, we made use of CosmoMC (Lewis & Bridle 2002; Lewis 2013). In this work, the threshold for the chains convergence measure, R − 1, was set to 0.025. Our statistical analysis and plotting are done by using Getdist (Lewis 2019).

For our purposes, we first split the CMB's Temperature-Temperature (TT) power spectrum into low and high ℓ's and check our model against them separately. This allows us to check if they are consistent in the GLT framework or not. If they are consistent then we are allowed to work with full CMB power spectrum data.

Here are the data combinations we have used:

• 1.  
  Planck 2018 CMB temperature power spectrum when ℓ ranges from 2 to 801. In addition, we have used the CMB low-ℓ polarization power spectrum, i.e., SimALL. We refer to this combination as low-ℓ (Planck Collaboration et al. 2020b).
• 2.  
  Planck 2018 CMB temperature power spectrum when ℓ ranges from 802 to 2500. We have also added SimALL to this combination because without which optical depth, τ, will not get constrained well. We refer to this combination as high-ℓ (Planck Collaboration et al. 2020b).
• 3.  
  Full Plank 2018 temperature and polarization power spectra and their cross correlations. We refer to this combination as P18 (Planck Collaboration et al. 2020b).
• 4.  
  P18, plus the power spectrum of the CMB lensing potential inferred from the four-point function of the CMB temperature map. We refer to this combination as P18+lensing (Planck Collaboration et al. 2020c).
• 5.  
  P18, plus BAO volume distance measurements, at z = 0.32 (LOWZ; Anderson et al. 2014), z = 0.57 (CMASS; Anderson et al. 2014), z = 0.106 (6dFGS; Beutler et al. 2011), and z = 0.15 (MGS; Ross et al. 2015). Furthermore we used BAO angular diameter distance measurements at z = 0.44, z = 0.60, and z = 0.73 (WiggleZ; Blake et al. 2011). We refer to this data combination as P18+BAO.

The set of parameters on its members we would like to put constraints on is ⁴

$$\begin{eqnarray}&&{ \mathcal P }=\{{{\rm{\Omega }}}_{b}{h}^{2},{{\rm{\Omega }}}_{c}{h}^{2},100{{\rm{\Theta }}}_{{MC}},\tau ,{n}_{s},\mathrm{ln}[{10}^{10}{A}_{s}],{a}_{t},\mathrm{ln}(\gamma ),{A}_{L}\}.\end{eqnarray} \tag{ 20 }$$

The first six parameters are common with ΛCDM and A_L can be thought of as an extension to it. But a_t and $\mathrm{ln}(\gamma )$ are two extra free parameters in GLT. In our analysis, we prefer to work with ${a}_{t}\equiv \tfrac{1}{1+{z}_{t}}$, instead of z_t , since equations and functions in CAMB are written in terms of scale factor.

In Table 1, the details about priors on these parameters and the data combination we have used for each model can be found.

Zoom In Zoom Out Reset image size

Table 1. Parameters vs. Models

Parameter	Prior	ΛCDM (P18)	GLT (P18)	ΛCDM+AL (P18)	GLT+AL (P18)	ΛCDM+AL (P18+lensing)	GLT+AL (P18+lensing)	GLT+AL (P18+BAO)
Ωb h2	[0.005, 0.1]	0.02236 ± 0.00015	0.02259 ± 0.00016	0.02259 ± 0.00017	0.02261 ± 0.00017	0.02248 ± 0.00016	0.02260 ± 0.00017	0.02252 ± 0.00015
Ωc h2	[0.001, 0.99]	0.1202 ± 0.0014	0.1179 ± 0.0015	0.1181 ± 0.0016	0.1178 ± 0.0016	0.1185 ± 0.0015	0.1178 ± 0.0016	0.1188 ± 0.0011
100ΘMC	[0.5, 10]	1.04090 ± 0.00031	1.04119 ± 0.00032	1.04114 ± 0.00032	1.04119 ± 0.00033	1.04112 ± 0.00032	1.04124 ± 0.00033	1.04108 ± 0.00030
τ	[0.01, 0.8]	${0.0544}_{-0.0081}^{+0.0070}$	${0.0500}_{-0.0071}^{+0.0082}$	${0.0492}_{-0.0073}^{+0.0088}$	${0.0487}_{-0.0072}^{+0.0092}$	${0.0486}_{-0.0076}^{+0.0090}$	${0.0492}_{-0.0075}^{+0.0088}$	0.0491 ± 0.0082
ns	[0.8, 1.2]	0.9649 ± 0.0044	0.9712 ± 0.0048	0.9708 ± 0.0048	0.9716 ± 0.0049	0.9692 ± 0.0049	0.9716 ± 0.0049	0.9691 ± 0.0041
$\mathrm{ln}[{10}^{10}{A}_{s}]$	[2, 4]	3.045 ± 0.016	${3.030}_{-0.015}^{+0.017}$	${3.029}_{-0.015}^{+0.018}$	${3.027}_{-0.016}^{+0.019}$	${3.028}_{-0.016}^{+0.019}$	${3.028}_{-0.016}^{+0.018}$	${3.030}_{-0.016}^{+0.018}$
at	[0.01, 1]	⋯	<0.329	⋯	<0.189	⋯	<0.190	<0.171
$\mathrm{ln}(\gamma )$	[−16, 16]	⋯	$-{6.8}_{-13}^{+4.6}$	⋯	not constrained	⋯	<−2.65	not constrained
AL	[0.5, 2]	⋯	⋯	1.180 ± 0.065	${1.072}_{-0.11}^{+0.084}$	${1.051}_{-0.041}^{+0.037}$	${1.002}_{-0.044}^{+0.038}$	${1.062}_{-0.11}^{+0.078}$
H0 [km/s/Mpc]	⋯	67.27 ± 0.60	${70.42}_{-2.4}^{+0.94}$	68.28 ± 0.72	${70.41}_{-2.2}^{+0.96}$	68.05 ± 0.70	${70.44}_{-2.3}^{+0.93}$	${68.69}_{-0.68}^{+0.55}$
Total ${\chi }_{\min }^{2}$	⋯	2772.6	2764.8	2764.1	2763.5	2789.4	2780.0	2768.1
${\chi }_{{\rm{P}}18}^{2}$	⋯	2768.9	2759.6	2760.8	2760.5	2777.4	2769.1	2761
${\chi }_{\mathrm{lensing}}^{2}$	⋯	⋯	⋯	⋯	⋯	9.2	7.8	⋯
${\chi }_{\mathrm{BAO}}^{2}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	4.6

Note. The confidence regions here are 68%. One would infer that ΛCDM is not consistent with its own prediction about the amplitude of lensing unless we include lensing data. GLT, on the other hand, is able to have A_L = 1 even when lensing data is not included. The last four rows compare the models with the measure of χ². As can be seen, our model is favored over ΛCDM when confronting P18 alone. This is because the lensing effect is naturally enhanced and at high ℓ's the theory has a better fit to data. When we add A_L to the models, fluctuations of dark energy are not as useful as before and ΛCDM and GLT have more or less the same goodness of fit as P18. However, this goodness of fit for ΛCDM comes at the cost of A_L ≠ 1, while GLT provides the possibility of being unity for A_L . This would get more apparent by adding lensing data since lensing data prefers A_L = 1. In the last column, while the BAO data has made the H₀ get close to what ΛCDM predicts, A_L = 1 has remained inside the 1σ confidence region. The absent row above is ${\chi }_{\mathrm{prior}}^{2}$, which CosmoMC reports, but we did not include it in the table.

Download table as:  ASCIITypeset image

In Figure 4, we have shown that the 1σ contours of low and high ℓ's are overlapped. This means they are consistent with each other in the GLT framework unlike ΛCDM. Hence it is statistically meaningful to combine these two data sets in the framework of GLT.

Now it is time to consider the CMB lensing anomaly. In the literature, as is also mentioned by Planck Collaboration et al. (2020a), the CMB-lensing and low/high-ℓ inconsistencies are suggested to be related theoretically. The reason is that it is expected that the lensing potential affects high ℓ's (small scales) more than low ℓ's (large scales) in ${C}_{\mathrm{TT}}^{{\ell }}$'s. To check this proposal, the lensing amplitude A_L became relaxed to see if our GLT model can relieve the CMB lensing anomaly. In Figure 2, we have shown that the fluctuations in GLT can indeed address the CMB lensing anomaly completely. More quantitatively, in Table 1, it can be seen that while in the ΛCDM+A_L model, the A_L = 1 is 2.8σ away from the best fit, in GLT+A_L , it is within 1σconfidence level. This is important because lensing data itself prefers A_L ≈ 1 and its inclusion pushes A_L toward unity by force and raises the χ² significantly. Note that this implicitly suggests that it is not statistically valid to combine P18 and lensing data in ΛCDM. In GLT though, A_L = 1 is already accessible and the addition of lensing data does not penalize the model that much.

Zoom In Zoom Out Reset image size

On the other hand, the temporal evolution of the dark energy density that is described in Equation (6), is able to ease the discrepancy between the derived H₀ from CMB and local measurements that are higher. Note that if we consider BAO's then there is no chance to address the H₀ tension but still we address the internal inconsistencies, as can be seen in Figure 2 and Figure 3.

Zoom In Zoom Out Reset image size

In this work, we studied the Ginzburg–Landau theory of dark energy (GLT) beyond its mean-field approximation. The GLT is based on several assumptions: dark energy is somehow sensitive to the cooling down of the universe and hence undergoes a phase transition, the evolution of dark energy comes from the so-called Landau free energy, and any details about both temporal or spatial features of this dark energy can be deduced from this effective free energy. While in previous works (Banihashemi et al. 2019, 2020, 2021), we studied GLT at the background level without quantitative considerations of its spatial fluctuations, this work is our first attempt to see the fingerprints of GLT in its very specific spatial properties. This can be seen as the smoking gun for our GLT model since it is very different from other dark energy models, e.g., quintessence. It turned out that the time and scale dependencies of the dark energy patches in this scenario are such that ease the low- and high-ℓ inconsistency in the CMB angular power spectrum (it can be checked qualitatively in Figure 4) and also solves the CMB lensing anomaly (see Figure 2 and Table 1 for discussion). The addition of BAO data prevents GLT from fully solving the H₀ tension, but we would like to emphasize that while GLT fluctuations can justify the internal inconsistencies in Planck data and also solve the lensing anomaly, they will not worsen the Hubble tension at all. Note that relaxing Ω_k in the ΛCDM framework does the same job, but on the other hand extremely intensifies the H₀ tension and also makes the BAO in strong disagreement with the predictions (Di Valentino et al. 2020).

Zoom In Zoom Out Reset image size

There are also other peculiar features in GLT that lie beyond the scope of this work. As an example, this model predicts a transient longwave mode in dark energy density and this may challenge the isotropy of the universe.

In the future, we not only plan to search for other observational implications of the GLT, but also we wish to build it on a more concrete theoretical ground with less phenomenological bases.

A.B. is in debt to the Institute for Theoretical Physics at the University of Heidelberg, where this work was initiated, for their hospitality. We would like to thank Luca Amendola for his insightful suggestions and comments during our talks on GLT. We thank Iran National Science Foundation (INSF), for supporting this work under project No. 98022568.