CMB Parity Asymmetry from Unitary Quantum Gravitational Physics

Abstract

Longstanding anomalies in the Cosmic Microwave Background (CMB), including the low quadrupole moment and hemispherical power asymmetry, have recently been linked to an underlying parity asymmetry. We show here how this parity asymmetry naturally arises within a quantum framework that explicitly incorporates the construction of a geometric quantum vacuum based on parity ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) transformations. This framework restores unitarity in quantum field theory in curved spacetime (QFTCS). When applied to inflationary quantum fluctuations, this unitary QFTCS formalism predicts parity asymmetry as a natural consequence of cosmic expansion, which inherently breaks time-reversal symmetry. Observational data strongly favor this unitary QFTCS approach, with a Bayes factor, the ratio of marginal likelihoods associated with the model given the data $p\left(M|D\right)$, exceeding 650 times that of predictions from the standard inflationary framework. This Bayesian approach contrasts with the standard practice in the CMB community, which evaluates $p\left(D|M\right)$, the likelihood of the data under the model, which undermines the importance of low-ℓ physics. Our results, for the first time, provide compelling evidence for the quantum gravitational origins of CMB parity asymmetry on large scales.

1. Introduction

Understanding and testing the interplay between gravity and quantum mechanics (QM) is the most important key to unraveling the new physics beyond the standard model of particle physics and Einstein’s general relativity (GR). The merger of GR and QM is an essential step in physics that can be unlocked at the length scales of gravitational horizons. For this, we perhaps need an in-depth understanding of how one can resolve an age-old conundrum in the concept of time between GR and QM. The success of quantum field theory (QFT) in Minkowski spacetime [1] already teaches us several lessons in this regard. Indeed, QFT is the culmination of special relativity (SR) and QM, where the concept of time already enters into the conflict. In SR, space and time are on equal footing, but in QM, spatial coordinates are elevated to operators while time is just a parameter due to its anti-unitary nature under the discrete transformation ($\mathcal{T}:t\to -t$). SR and QM are merged by imposing the commutativity of operators corresponding to space-like distances. The underlying reason is that SR forbids any communication happening in space-like distances, and we respect this and achieve this by formulating new mathematical rules via second quantization. This is exactly the lesson one must keep in mind to understand quantum fields in curved spacetime. That is, we take lessons from classical physics and comply with them according to the rules of quantum physics. Quantum field theory in curved spacetime (QFTCS) is the first step in seeing GR and QM acting together, besides the renormalizable ultra-violet complete quantum gravity that may be required at the Planck scale. Furthermore, QFTCS is a necessity in the context of understanding inflationary quantum fluctuations [2,3,4]. The longstanding problem of QFTCS has been the loss of unitarity and the information paradox, which was first formulated in the context of black holes by Hawking’s seminal works [5,6] and later extended to de Sitter space in the paper of Gibbons and Hawking [7]. The problem of unitarity emerges with the way quantization is usually achieved in curved spacetime, leading to the phenomenon of pure states evolving into mixed states. This means that an observer bounded by a spacetime horizon loses access to the states beyond the horizon. This problem was first identified by Schrödinger in 1956 in the context of de Sitter space [8]. Despite some attempts in the setups of de Sitter holography [9,10] that attempted to save unitarity, multiple investigations in recent decades have largely focused on accepting unitarity loss and proposing entangled pairs of universes and parallel spacetimes or restoring unitarity from ad hoc yet unknown frameworks of quantum gravity (see [11,12,13,14,15,16,17] and references therein). Motivated by ’t Hooft, Schrödinger, and Einstein–Rosen’s seminal works [8,18,19], which insisted on the role of parity and time-reversal operations, the recent proposal of direct-sum quantum field theory (DQFT) restores the unitarity in curved spacetime [20,21,22,23,24] along with a potential solution to the information loss paradox.

In this paper, we address crucial questions related to the meaning of time-reversal and parity inversions in the context of inflationary quantum fluctuations and uncover their true nature in the cosmic microwave background (CMB). We explain how DQFT [20,21,22]) is a necessity to solve the unitarity problem of QFTCS. The question of unitarity in QFTCS precedes the question of a completely renormalizable quantum gravity theory at Planck scales. In contrast to the widespread schools of thought that argue unitarity in QFTCS needs to be given up within GR and QM, we propose a new fundamental understanding of this issue and a new observational test using the parity asymmetry in the CMB. We find that the theory of inflationary quantum fluctuations with DQFT (which we abbreviate to DSI, which stands for “Direct-Sum Inflation”) fits up to 650 times better than Standard Inflation (SI). This paper ultimately presents a picture of how a new theory of QFTCS emerged from fundamental questions of quantum gravitational physics that can potentially explain the observed CMB (large-scale) anomalies. The new results in this paper are further complemented by our more extended companion paper [25].

This paper is organized as follows: In Section 2 we summarize the formulation of DQFT, which is built on a new understanding of quantum theory with two arrows of time along with parity-based geometric superselection rules. In Section 3, we discuss the construction of DQFT in de Sitter spacetime and demonstrate how we achieve unitarity in this setup. In Section 4, we present the application of DQFT to inflationary quantum fluctuations (i.e., DSI) and provide strong statistical evidence for it over the standard framework of inflation (SI) using the latest CMB data from the Planck scale. In Section 5 we conclude with the important elements of our investigation, which expose the role of gravity and quantum mechanics in the early universe and the fresh perspective we bring to the subject with compelling observational evidence. In Appendix A, Appendix B and Appendix C, we provide further essential details of direct-sum QM and DQFT in Miknowski and de Sitter spacetimes.

2. DQFT, in a Nutshell ($\hslash =\mathbf{c}=\mathbf{1}$)

The formulation of DQFT emerges from a simple reconstruction of a quantum state or a quantum field operator through the discrete (a) symmetries (parity $\mathcal{P}$ and time-reversal $\mathcal{T}$) of the background spacetime. Here, we summarize the conceptual understanding of DQFT and direct the reader to [20,21,22,23,24,25,26] for more details. Construction of QFT in Minkowski spacetime requires the definition of a positive energy state and the arrow of time [27]. Here, we build quantum theory with two arrows of time using the direct-sum Schrödinger equation:

$$i{\displaystyle \frac{\partial |\Psi \rangle }{\partial t_p}}=\left(\begin{array}{cc}{\widehat{\mathbb{H}}}_{+}& 0\\ 0& -{\widehat{\mathbb{H}}}_{-}\end{array}\right)|\Psi \rangle$$

(1)

where

$$\widehat{\mathbb{H}}\left(\widehat{x},\widehat{p}\right)={\widehat{\mathbb{H}}}_{+}\left({\widehat{x}}_{+},{\widehat{p}}_{+}\right)\oplus {\widehat{\mathbb{H}}}_{-}\left({\widehat{x}}_{-},{\widehat{p}}_{-}\right)=\left(\begin{array}{ccc}{\widehat{\mathbb{H}}}_{+}& & 0\\ 0& & {\widehat{\mathbb{H}}}_{-}\end{array}\right)$$

(2)

is the time-independent Hamiltonian operator (which is Hermitian), which we assume here to be $\mathcal{P}\mathcal{T}$-symmetric for simplicity It is important to note that our construction is not related to $\mathcal{P}\mathcal{T}$-symmetric QM where systems with non-Hermitian Hamiltonians are studied [28]. In contrast, we work here with Hermitian Hamiltonian operators. The position operator is split into a direct-sum of two components (${\widehat{x}}_{\pm }$) as

$$\widehat{x}={\widehat{x}}_{+}\oplus {\widehat{x}}_{-}=\left(\begin{array}{ccc}{\widehat{x}}_{+}& & 0\\ 0& & {\widehat{x}}_{-}\end{array}\right)$$

(3)

which have parity conjugate eigenvalues when they act on the states $|{\Psi }_{\pm }\rangle$. Similarly, the momentum operator is split as

$$\widehat{p}={\widehat{p}}_{+}\oplus {\widehat{p}}_{-}=\left(\begin{array}{ccc}{\widehat{p}}_{+}& & 0\\ 0& & {\widehat{p}}_{-}\end{array}\right)$$

(4)

with ${\widehat{p}}_{\pm }$ as the momentum operators associated with the states $|{\Psi }_{\pm }\rangle$ that correspond to Hilbert spaces representing parity conjugate regions of physical space as explained further below. The quantum state $|\Psi \rangle$ in Equation (1) is expressed as the direct-sum of two components:

$$\begin{array}{ccc}\hfill |\Psi \rangle & =& {\displaystyle \frac{1}{\sqrt{2}}}\left(|{\Psi }_{+}\rangle \oplus |{\Psi }_{-}\rangle \right)={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}|{\Psi }_{+}\rangle \\ |{\Psi }_{-}\rangle \end{array}\right)\hfill \\ \hfill {\int }_{-\infty }^{\infty }\langle \Psi |\Psi \rangle dx& =& {\displaystyle \frac{1}{2}}{\int }_{-\infty }^{\infty }\left[\langle {\Psi }_{+}|{\Psi }_{+}\rangle +\langle {\Psi }_{-}|{\Psi }_{-}\rangle \right]dx=1.\hfill \end{array}$$

(5)

where $|{\Psi }_{+}\rangle$ and $|{\Psi }_{-}\rangle$ are the two components of the same state at parity conjugate points. The component $|{\Psi }_{+}\rangle$ is a positive-energy state ($\mathcal{E}>0$) that evolves forward in time as $|{\Psi }_{+}{\rangle }_{t_p}=e^{-i\mathcal{E}{t}_p}{|{\Psi }_{+}\rangle }_0$ where the parametric time ($t_p$) arrow $t_p:-\infty \to \infty$ while $|{\Psi }_{-}\rangle$ can also be seen as a positive energy state that evolves back in time as $|{\Psi }_{-}{\rangle }_{t_p}=e^{i\mathcal{E}{t}_p}{|{\Psi }_{-}\rangle }_0$ with the parametric time convention $t_p:+\infty \to -\infty$. One can notice here the purely antiunitary character of time associated with the replacement $i\to -i$ [27]. Note that the minus sign in front of ${\mathbb{H}}_{-}$ in Equation (1) represents the opposite arrow of time. Thus, the direct-sum Schrödinger equation does not have any arrow of time associated with it compared to the conventional Schrödinger equation. To be explicit, we can rewrite Equation (1) as

$$\left(\begin{array}{c}i{\displaystyle \frac{\partial }{{\partial }_{t_p}}}|{\Psi }_{+}\rangle \\ -i{\displaystyle \frac{\partial }{{\partial }_{t_p}}}|{\Psi }_{-}\rangle \end{array}\right)=\left(\begin{array}{ccc}{\widehat{\mathbb{H}}}_{+}& & 0\\ 0& & {\widehat{\mathbb{H}}}_{-}\end{array}\right)\left(\begin{array}{c}|{\Psi }_{+}\rangle \\ |{\Psi }_{-}\rangle \end{array}\right)$$

(6)

The direct-sum splitting of the quantum state as in Equation (5) implies the Hilbert space ($\mathcal{H}$) splitting into a direct-sum of the two (${\mathcal{H}}_{\pm }$) corresponding to parity conjugate points in position space: $\mathcal{H}={\mathcal{H}}_{+}\oplus {\mathcal{H}}_{-}$. Here, the Hilbert spaces ${\mathcal{H}}_{\pm }$ are called the geometric superselection sectors [20,22]. The position and momentum commutation relations now become double-associated with parity conjugate position and momentum space operators denoted by subscripts + and −, and their relations are

$$\begin{array}{cc}\hfill \left[\widehat{x},\widehat{p}\right]& =i\Rightarrow \left\{\begin{array}{c}\left[{\widehat{x}}_{+},{\widehat{p}}_{+}\right]=-\left[{\widehat{x}}_{-},{\widehat{p}}_{-}\right]=i\hfill \\ [{\widehat{x}}_{+},{\widehat{x}}_{-}]=[{\widehat{p}}_{+},{\widehat{p}}_{-}]=[{\widehat{x}}_{\pm },{\widehat{p}}_{\mp }]=0\hfill \end{array}\right.\hfill \\ \hfill {\widehat{p}}_{\pm }& =\mp i{\displaystyle \frac{\partial }{\partial x_{\pm }}},\begin{array}{cc}\hfill x_{+}& =x\in (0,\infty ]\hfill \\ \hfill x_{-}& =x\in [-\infty ,0)\hfill \end{array}\hfill \end{array}$$

The concept of the wavefunction in the direct-sum QM becomes

$$\Psi \left(x\right)={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}\langle x_{+}|\langle x_{-}|\end{array}\right)\left(\begin{array}{c}|{\Psi }_{+}{\rangle }_0{e}^{-i\mathcal{E}{t}_p}\\ |{\Psi }_{-}{\rangle }_0{e}^{i\mathcal{E}{t}_p}\end{array}\right)\Rightarrow \left\{\begin{array}{c}{\displaystyle \frac{1}{\sqrt{2}}}{\Psi }_{+}\left(x_{+}\right)e^{-i\mathcal{E}{t}_p},x_{+}=x\gtrsim 0\hfill \\ {\displaystyle \frac{1}{\sqrt{2}}}{\Psi }_{-}\left(x_{-}\right)e^{i\mathcal{E}{t}_p},x_{-}=x\lesssim 0.\hfill \end{array}\right.$$

(7)

This framework makes the wavefunction of the quantum system explicitly $\mathcal{P}\mathcal{T}$-symmetric that resonates with the assumed $\mathcal{P}\mathcal{T}$ symmetry of the physical system—i.e., $[\mathbb{H},\mathcal{P}\mathcal{T}]=0$ and the results of standard quantum mechanics remain the same as shown with the example of the harmonic oscillator worked out in [25] (see also further discussions and Figure  1 in [20,23]). The DQFT in Minkowski spacetime: $d{s}^2=-d{t}_m^2+d{\mathbf{x}}^2$, which is $\mathcal{P}\mathcal{T}$-symmetric under $t_m\to -t_m$ and $\mathbf{x}\to -\mathbf{x}$, is just built on the direct-sum Schrödinger equation given in Equation (1), following how the standard QFT is built on the definition of a positive-energy state and the causality condition (i.e., commutativity of operators for spacelike distances). See Appendix B for more details.

3. DQFT in de Sitter and Unitarity ($\mathbf{G}=\hslash =\mathbf{c}=\mathbf{1}$)

De Sitter (dS) spacetime is a perfect testing ground for exploring the theory of QFTCS because of its maximally symmetric aspect, and it is also relevant for unlocking the nature of inflationary quantum fluctuations. Furthermore, dS spacetime has the closest resemblance to black holes, where the most important conundrums in quantum physics emerge, such as the loss of unitarity and the information loss paradox [7]. We start with the dS metric in the flat Friedman–Lemaître–Robertson–Walker (FLRW) coordinates:

$$d{s}^2=-d{t}^2+e^{2Ht}d{\mathbf{x}}^2={\displaystyle \frac{1}{H^2{\tau }^2}}\left(-d{\tau }^2+d{\mathbf{x}}^2\right),$$

(8)

where the scale factor $a=e^{Ht}$ is the clock that determines the expansion of the universe, $H={\displaystyle \frac{\dot{a}}{a}}$ is the Hubble parameter, and $\tau =\int {\displaystyle \frac{dt}{a}}=-{\displaystyle \frac{1}{aH}}$ is the conformal time for $a=e^{Ht}$. If one has to place quantum theory in dS spacetime as given in Equation (8), the first and foremost things to be understood are the parity and time-reversal operations. The usual conception of understanding the expansion of the universe is to restrict ourselves to $\tau <0$. But the metric carries a symmetry as follows:

$$\mathcal{T}:\tau \to -\tau ,\mathcal{P}:\mathbf{x}\to -\mathbf{x}$$

(9)

Restricting to $\tau <0$ before quantization of a classical field (as is usually followed in standard QFTCS) means throwing away the symmetry in Equation (9) by hand. The expanding universe can be described in two ways:

$$\begin{array}{cc}\hfill & \Rightarrow \left\{\begin{array}{cc}t:-\infty \to +\infty ,H>0\hfill & (\tau :-\infty \to 0)\hfill \\ t:+\infty \to -\infty ,H<0\hfill & (\tau :\infty \to 0)\hfill \end{array}\right.\hfill \end{array}$$

(10)

Notice in particular that in Equation (10), reversing the arrow of time together with the sign of the Hubble parameter represents the same expanding universe. This means that the scale factor is the clock that determines the expansion of the universe. This would explain why in the Wheeler–de Witt equation the cosmic time does not explicitly appear; rather, the scale factor is the clock that runs with the expanding universe [29]. Note that $H\to -H$ does not change any properties of the dS space at all. For example, curvature invariants such as the Ricci scalar $R=12H^2$ remain the same under a discrete sign flip of H. Equation (10) illustrates two possible time realizations for an expanding universe. By convention, choosing $\tau >0$ or $\tau <0$ results in loss of information beyond the horizon. In his famous monograph of 1956, Schrodinger [8] rejected the idea of two universes because it would lead to a conundrum of ignorance of information beyond the horizon, which in modern language means pure states evolving into mixed states, which leads to the violation of unitarity and information loss [7,9].

Similar to the Minkowski space (see Appendix B), the quantum fields in the dS space are described through a direct-sum vacuum ${|0\rangle }_{\mathrm{dS}}=|0_{+}{\rangle }_{\mathrm{dS}}\oplus {|0_{-}\rangle }_{\mathrm{dS}}$ corresponding to the direct-sum Fock space ${\mathcal{F}}_{\mathrm{dS}}={\mathcal{F}}_{\mathrm{dS}+}\oplus {\mathcal{F}}_{\mathrm{dS}-}$. This implies that everywhere in dS space, a quantum field is tied by time forward and backward evolutions at the parity conjugate points; see Appendix C. At any moment of dS expansion, the parity conjugate points on the comoving horizon $r_H=\left|{\displaystyle \frac{1}{aH}}\right|$ are space-like separated for an imaginary classical observer as shown in Figure 1. Note that by DQFT construction, the direct-sum quantum states $|{\varphi }_{\pm }\rangle ={\widehat{d}}_{(\pm )\mathbf{k}}^{†}{|0_{\pm }\rangle }_{\mathrm{dS}}$ that follow from Equation (A14) are the opposite time-evolving positive-energy states at the parity conjugate points, and this holds for any imaginary classical observer. Furthermore, in DQFT, causality and locality are preserved too. A consequence of this is that any state that disappears beyond the horizon (say at $\left(\theta ,\phi \right)$) happens to be the complementary direct-sum (component) state that reappears at the antipodal point (at $\left(\pi -\theta ,\pi +\phi \right)$). This is because all the imaginary classical observers, including those on the horizons of each other, should observe the same direct-sum quantum state $|{\varphi }_{+}\rangle \oplus |{\varphi }_{-}\rangle$.

This means the horizon acts as a $\mathcal{P}\mathcal{T}$ mirror that maps every piece of information outside the horizon to the states inside the horizon. To see this, consider a pure state:

$$|\psi \rangle =\sum _m{d}_m|{\varphi }_m\rangle ,$$

(11)

where $|{\varphi }_m\rangle$ are the eigenvectors—i.e., for any observable (an operator) $\widehat{O}$, we have the following: $\widehat{O}|{\varphi }_m\rangle ={\lambda }_m|{\varphi }_m\rangle$ where ${\lambda }_m$ gives the corresponding eigenvalues. Each ${\lambda }_m$ can be observed in $|\psi \rangle$ with probability $P\left({\lambda }_m\right)={\left|d_m\right|}^2$. Consider now the (de Sitter) Hubble horizon $r_H=\left|{\displaystyle \frac{1}{aH}}\right|$ around an observer (as illustrated in Figure 1). In the standard approach, information is lost (a pure state becomes a mix state) because communication cannot happen for space-like distances (i.e., outside the horizon) as was shown by Gibbons and Hawking in 1977 [7], which is analogous to what happens in Hawking radiation and the black hole information loss paradox [5,21]. This also implies the loss of unitarity in QFT on spacetimes with horizons. In the DQFT approach, we have

$$|\psi \rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}\sum d_m^{+}|{\varphi }_m^{+}\rangle \\ \sum d_m^{-}|{\varphi }_m^{-}\rangle \end{array}\right)\equiv {\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}|{\psi }^{+}\rangle \\ |{\psi }^{-}\rangle \end{array}\right)$$

(12)

where $|{\varphi }_m^{\pm }\rangle$ are the eigenvectors corresponding to each $\mathcal{P}\mathcal{T}$-separated Hilbert space. So, the observable probabilities and operators become

$$P\left({\lambda }_m^{\pm }\right)={\left|d_m^{\pm }\right|}^2\mathrm{and}\widehat{O}=\left(\begin{array}{cc}{\widehat{O}}_{+}& 0\\ 0& {\widehat{O}}_{-}\end{array}\right)$$

(13)

Information is not lost in this case because there is no entanglement across the horizon. To see this more clearly, consider the case where $|\psi \rangle$ is a pure entangled system $|{\psi }_{AB}\rangle =|{\psi }_A\rangle \otimes |{\psi }_B\rangle$:

$$|{\psi }_{AB}\rangle =\sum _{m,n}{d}_{mn}|{\varphi }_{Am}\rangle \otimes |{\varphi }_{Bn}\rangle ,$$

(14)

where ⊗ represents the tensor product of two Hilbert spaces A and B that are entangled (inseparable or correlated): $d_{mn}\ne d_m{d}_n$, with $|{\varphi }_A\rangle ={\sum }_m{d}_m|{\varphi }_{Am}\rangle$ and $|{\varphi }_B\rangle ={\sum }_n{d}_n|{\varphi }_{Bn}\rangle$. For example, consider a pair of particles (across the horizon). In the DQFT approach, we have

$$|{\psi }_{AB}\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}\sum d_{mn}^{+}|{\varphi }_{Am}^{+}\rangle \otimes |{\varphi }_{Bn}^{+}\rangle \\ \sum d_{mn}^{-}|{\varphi }_{Am}^{-}\rangle \otimes |{\varphi }_{Bn}^{-}\rangle \end{array}\right)\equiv {\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}|{\psi }_{AB}^{+}\rangle \\ |{\psi }_{AB}^{-}\rangle \end{array}\right)$$

(15)

where $|{\varphi }_{Am}^{\pm }\rangle$ and $|{\varphi }_{Bn}^{\pm }\rangle$ are the eigenvectors corresponding to each $\mathcal{P}\mathcal{T}$-separated Hilbert space, which remain within the horizon because no information is exchanged across $\mathcal{P}\mathcal{T}$-separated spaces. The fact that there are no cross terms in Equation (15) follows from the definition of superselection sector Hilbert spaces, according to which one cannot form any superposition of states from the superselection sectors (see [30,31,32] for more details on superselection sectors and direct-sum Hilbert spaces and the associated tensor products). This is why in Equation (15) we do not have any cross-products between parity conjugate states. All information about the entanglement pairs is mapped within the horizon; thus, the pure state will remain pure. In other words, DQFT forbids any entanglement beyond the gravitational horizons. DQFT constructs a direct-sum Hilbert space with the interior and exterior separated by a geometric superselection rule (which is similar to the concept of superselection sectors in QFT [32,33]). Thus, there would be an observer complementarity (see Figure 3 in [20]). We can also understand the same from the point of view of dS space in static coordinates, which reads as

$$\begin{array}{cc}\hfill d{s}^2& =-\left(1-H^2{r}^2\right)d{t}_s^2+{\displaystyle \frac{1}{\left(1-H^2{r}^2\right)}}d{r}^2+r^{2}d{\Omega }^2\hfill \\ \hfill & ={\displaystyle \frac{1}{H^2{\left(1-UV\right)}^2}}\left(-4dUdV+{\left(1+UV\right)}^{2}d{\Omega }^2\right)\hfill \end{array}$$

(16)

where $r=\left|{\displaystyle \frac{1}{H}}\right|{\displaystyle \frac{1+UV}{1-UV}}$ and ${\displaystyle \frac{V}{U}}=-e^{2H{t}_s}$. The relation between the flat FLRW Equation (8) and Equation (16) is just a simple coordinate change [34,35]. A widely practiced view of flat dS spacetime in Equation (8) is that it covers only half of the dS space. However, we must keep in mind the symmetry in Equation (10), which implies that Equation (8) can cover the entire dS space. Thus, the maximally symmetric nature of the dS spacetime makes no difference whether it is expressed in flat, closed, or open FLRW coordinates, or static coordinates [20,36]. The use of mapping flat coordinates to static coordinates here is that we can realize how unitarity can be achieved with DQFT, whereas it is lost in standard QFT, as was shown by Gibbons and Hawking in 1977 [7]. In Figure 2, we show unitary quantum physics in the static dS space through a (quantum) conformal diagram, where we see the three-dimensional space divided by parity for the interior and exterior of the cosmological event horizon. We can picture here how none of the observers loses any information beyond the horizon. If any state leaves the horizon, it is only the (complimentary) direct-sum component of what is inside the horizon. This means that any observer can perfectly reconstruct the information about what is beyond the horizon. According to DQFT, every density matrix of a maximally entangled pure state is split into the direct-sum of four components:

$$\rho =\left({\rho }_I\oplus {\rho }_{II}\right)\oplus \left({\rho }_{III}\oplus {\rho }_{IV}\right)$$

(17)

Each density matrix ${\rho }_n,n=I,II,III,IV$ is defined by a pure state component in the corresponding (geometric) superselection sector Hilbert space ${\mathcal{H}}_n$. Since the Von Neumann entropy of each component of the density matrix vanishes, $S_n=-Tr{\rho }_{n}ln{\rho }_n=0$, any observer in regions I, II, III, and IV (see Figure 2) would witness pure states evolving into pure states. Thus, with DQFT, we can have both unitarity and observer complementarity in dS [20]. Furthermore, this DQFT construction is also studied in the context of achieving unitarity in Rindler spacetime [26].

According to DQFT, any state is spread across the horizon by distinct components in geometric superselection sectors. This implies that any information of the entangled state is shared by the sectorial Hilbert space, leading to pure states evolving into pure states. To be precise, the Hilbert spaces of the interior and exterior states cannot be connected by direct-product (⊗) but rather should be related by direct-sum (⊕). This has to be like this because beyond-the-horizon time is not the same, so one cannot apply the same quantum mechanics everywhere. In other words, the gravitational horizons are special; they separate different spacetimes by a boundary. Thus, they cannot be treated as a standard null surface, across which quantum states can be entangled. Furthermore, we stress that no entanglement across the horizon does not lead to firewalls, which happens in standard QFTCS but not in direct-sum QFTCS. For further analogous discussion on unitarity and no information loss in the context of the Schwarzschild black hole, see [21].

In a nutshell, DQFT offers a new understanding of dS space where any observer’s physics of the universe is complete, and no information is lost outside any observer’s universe. This is exactly what Schrödinger envisioned in 1956 for a quantum theory to achieve [8].

The formalism of DQFT has certain similarities with ’tHooft’s recent ideas on black holes [18]. However, there are some crucial differences, one of them being the construction of the geometric superselection rule that restructures a single quantum state in sectorial Hilbert spaces that represent parity conjugate regions of physical space. ’tHooft on the other hand considers anti-podal identification of time-reversal regions of spacetime with a slightly different interpretation of parity. The idea of antipodal identification was later given up by ’t Hooft because of the possibility of closed timeline curves when dynamics of spacetime are involved, so a new idea of quantum clones was proposed [37]. Our framework significantly differs from this latter proposal; we speak about $\mathcal{P}\mathcal{T}$-based mirror symmetries within the description of quantum fields in the same manifold rather than creating clones in different manifolds. One can also notice some aesthetic similarities between our proposal and Boyle and Turok’s CPT symmetric universes [38,39] (coincidentally, it is a new understanding of the old proposal by Sakharov [40,41]). The crucial difference in our framework is that we are speaking about parity and time-reversal within the same expanding universe, in contrast with Boyle and Turok’s pair of CPT-symmetric universes before and after the Big Bang. Furthermore, DQFT echoes in parallel the recent classical understanding of dS through the Black Hole Universe (BHU) proposal [42,43,44].

4. CMB Parity Asymmetry

Inflationary quantum fluctuations are the initial seeds for large-scale structure formation. It is important to understand their quantum nature to derive consistent CMB predictions. Inflation is by definition a quasi-de Sitter expansion [45] and this means that $\mathcal{P}\mathcal{T}$ in Equation (10) is spontaneously broken by the presence of a non-perturbative scalar field in addition to the GR’s tensor degree of freedom. Our focus here is on single-field inflation. The metric and scalar field matter quantum fluctuations are generated during inflation. The quantity that is a gauge-invariant combination of them is called the curvature perturbation ($\zeta$), and it is an effective (scalar) degree of freedom ($\varphi$). It connects the quantum physics of inflation to CMB observations. The perturbed action for curvature perturbation at the linear order is the following [25,46]:

$${\delta }^{\left(2\right)}{S}_s={\displaystyle \frac{1}{2}}\int d\tau d^{3}x{a}^2{\displaystyle \frac{{\dot{\overline{\varphi }}}^2}{H^2}}\left[{\zeta }^{\prime 2}-{\left({\partial }_i\zeta \right)}^2\right],$$

(18)

where $H={\displaystyle \frac{\dot{a}}{a}}$, which is nearly constant during inflation characterized by the smallness of slow-roll parameters $ϵ=-{\displaystyle \frac{\dot{H}}{H^2}}$ and $\eta ={\displaystyle \frac{\dot{ϵ}}{Hϵ}}$ for N = 50–60 (number of e-foldings). In Equation (18), the overbar indicates the background field, overdot indicates time derivative, ′ denotes derivative with respect to conformal time, and ${\partial }_i={\displaystyle \frac{\partial }{\partial x^i}}$ implies spatial gradient. The canonical variable that is a redefinition of curvature perturbation that we eventually quantize is

$$v=a\zeta \dot{\overline{\varphi }}/H$$

(19)

In the framework of direct-sum inflation (DSI), the canonical variable is promoted to a field operator that is split into direct-sum of two components:

$$\begin{array}{cc}\hfill \widehat{v}& ={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{cc}{\widehat{v}}_{+}\left(\tau ,\mathbf{x}\right)& 0\\ 0& {\widehat{v}}_{-}\left(-\tau ,-\mathbf{x}\right)\end{array}\right)\hfill \end{array}$$

(20)

which can be expanded as

$$\begin{array}{cc}\hfill & {\widehat{v}}_{\pm }=\int {\displaystyle \frac{d^{3}k}{{\left(2\pi \right)}^{3/2}}}\left[c_{\left(\pm \right)\mathbf{k}}{v}_{\pm ,k}{e}^{\pm i\mathbf{k}·\mathbf{x}}+c_{\left(\pm \right)\mathbf{k}}^{†}{v}_{\pm ,k}^{*}{e}^{\mp i\mathbf{k}·\mathbf{x}}\right]\hfill \end{array}$$

(21)

This would also mean promoting the curvature perturbation as an operator in the following way, which involves classical rescaling:

$$\widehat{\zeta }={\displaystyle \frac{H}{a\dot{\overline{\varphi }}}}{|}_{\mathrm{classical}}\widehat{v}$$

(22)

These quantum fluctuations evolve in a direct-sum vacuum defined by

$${|0\rangle }_{\mathrm{qdS}}=|0_{+}{\rangle }_{\mathrm{qdS}}\oplus |0_{-}{\rangle }_{\mathrm{qdS}}=\left(\begin{array}{c}|0_{+}{\rangle }_{\mathrm{qdS}}\\ |0_{-}{\rangle }_{\mathrm{qdS}}\end{array}\right),c_{\pm \mathbf{k}}{|0_{\pm }\rangle }_{\mathrm{qdS}}=0.$$

(23)

in which they evolve forward and backward in time at the parity conjugate points in physical space, governed by the mode functions $v_{\pm ,k}$, which are the solutions of

$$v_{\pm ,k}^{\prime \prime }+\left(k^2-{\displaystyle \frac{{\nu }_s^{\left(\pm \right)2}-\frac{1}{4}}{{\tau }^2}}\right)v_{\pm ,k}^2=0,{\nu }_s^{\pm }\approx {\displaystyle \frac{3}{2}}\pm ϵ\pm {\displaystyle \frac{\eta }{2}}$$

(24)

Recall that the choice of mode functions determines the vacuum (see Appendix C for the discussion in the context of de Sitter). The solutions of (24) are

$$\begin{array}{cc}\hfill v_{\pm ,k}& \approx \sqrt{{\displaystyle \frac{1}{2k}}}{e}^{\mp ik\tau }\left(1\mp {\displaystyle \frac{i}{k\tau }}\right)\pm \left(ϵ+{\displaystyle \frac{\eta }{2}}\right){\displaystyle \frac{\sqrt{\pi }}{2\sqrt{k}}}\sqrt{\mp k\tau }{\displaystyle \frac{\partial H_{{\nu }_s^{\pm }}^{\left(1\right)}\left(\mp k\tau \right)}{\partial {\nu }_s^{\pm }}}{|}_{{\nu }_s^{\pm }=3/2}\hfill \\ & \equiv v_{\pm ,k}^{\mathrm{dS}}\left(1\pm \Delta v\right)\hfill \end{array}$$

(25)

where $H_{{\nu }_s}^{\left(1\right)}\left(z\right)$ gives the Hankel functions of the first kind and

$$v_{\pm ,k}^{\mathrm{dS}}=\sqrt{{\displaystyle \frac{1}{2k}}}{e}^{\mp ik\tau }\left(1\mp {\displaystyle \frac{i}{k\tau }}\right),\Delta v=\left(ϵ+{\displaystyle \frac{\eta }{2}}\right)\left[{\displaystyle \frac{1}{H_{3/2}^{\left(1\right)}\left(\frac{k}{k_{*}}\right)}}{\displaystyle \frac{\partial H_{{\nu }_s}^{\left(1\right)}\left(\frac{k}{k_{*}}\right)}{\partial {\nu }_s}}{|}_{{\nu }_s={\displaystyle \frac{3}{2}}}\right]$$

(26)

The mode functions of curvature perturbation can be expressed as follows:

$${\zeta }_{\pm k}={\displaystyle \frac{H}{a\dot{\overline{\varphi }}}}{v}_{\pm k}={\zeta }_{\pm k}^{\mathrm{dS}}\left(1\pm \Delta v\right);{\zeta }_{\pm k}^{\mathrm{dS}}\equiv {\displaystyle \frac{H}{a\dot{\overline{\varphi }}}}{v}_{\pm ,\mathbf{k}}^{\mathrm{dS}}$$

(27)

We can see here how the extra quasi-de Sitter contribution $\pm \Delta v$ is anti-symmetric in parity conjugate regions of space. By analogy with Equation (10), the time-reversal operation (quantum mechanically) in an expanding quasi-dS universe is (notation here is that any quantity $Q\to -Q$ means we replace the quantity by a negative sign of that quantity):

$$\tau \to -\tau \Rightarrow (t,H,ϵ,\eta )\to (-t,-H,-ϵ,-\eta ).$$

(28)

We stress that the above flip of the sign of the slow-roll parameters should not be thought of from a classical point of view. Indeed, if one does literally $t\to -t$ and $H\to -H$, the definition of slow-roll parameters does not change sign. However, what we have to keep in mind here is that quantum mechanically, the concept of time is different from the classical point of view. Time is a parameter in quantum theory and also in quantum field theory [20].

In quantum mechanics, coordinate time is not an operator but a parameter because of the fundamental reason that time-reflection is an anti-unitary operation [27,47,48,49]. It is worth recalling that the usual QFT in Minkowski spacetime, which is the unification of special relativity and quantum mechanics, is built on assuming an arrow of time $t_p:-\infty \to \infty$, a particular definition of a positive-energy ($\mathcal{E}>0)$ state, ${|\Psi \rangle }_t=e^{-i\mathcal{E}{t}_p}{|\Psi \rangle }_0$, and that the operators to commute for space-like distances. DQFT brings a new construction of quantum theory, which gives a new understanding of quantum physics with both arrows of time.

When it comes to gravity, we encounter the dynamical nature of spacetime, so when we speak about quantum fields in curved spacetime, not only time but also any additional quantities that define the dynamics of spacetime act as parameters quantum mechanically. When we study inflationary fluctuations, which are a combination of metric and matter degrees of freedom (indeed the curvature perturbation $\zeta =-\Psi +H{\displaystyle \frac{\delta \varphi }{\dot{\overline{\varphi }}}}$ is a combination of Bardeen potential $\Psi$ and the inflation field fluctuation $\delta \varphi$), what we apply is linearized quantum gravity (see [50] for more discussion), which is encapsulated by the following equation

$$\delta {\widehat{G}}_{\mu \nu }=\delta {\widehat{T}}_{\mu \nu },$$

(29)

which also includes tensor fluctuations. In this paper, we only address the parity asymmetry in two-point functions of scalar fluctuations that leads to asymmetry in the even–odd angular scalar power spectra that are significant on large scales) according to the DSI model, but the DSI also predicts parity asymmetry in the tensor two point correlations as explained in [26].

The above sign changes (once more interpreted quantum mechanically) in the slow-roll parameters, which break the time-reversal symmetry of de Sitter spacetime, accommodate a new vacuum structure in quasi-de Sitter spacetime that lead to corrections in the evolution of quantum fluctuations ${\widehat{v}}_{(\pm )}$ in the parity conjugate regions. In the case of exact de Sitter spacetime, the DQFT framework leads to equal correlations for field components in the $\mathcal{P}\mathcal{T}$ conjugate regions, but in the quasi-de Sitter we obtain unequal correlations because the formulation of our quasi-de Sitter vacuum Equation (23) leads to fluctuations evolving asymmetrically at parity conjugate points in physical space. Thus, it is a pure quantum (gravitational) effect, and it has no classical analog. One can interpret this result as inflation breaks spontaneously the $\mathcal{P}\mathcal{T}$ symmetry of de Sitter space, implying the quantum fluctuations evolve asymmetrically at parity conjugate points (see Figure 3 in [25] or Figure 1 above). As we will show, this results in an asymmetry in the CMB temperature under spatial parity. The two (scalar) power spectra defined with respect to the two quasi-de Sitter vacua in Equation (23) correspond to parity conjugate regions of physical space. From Equation (27) we have the following:

$${\mathcal{P}}_{\zeta \pm }=\int {d^{3}r{e}^{-i\mathbf{k}.\mathbf{r}}}_{\mathrm{qdS}}\langle 0_{\pm }|{\widehat{\zeta }}_{\pm }\left(\pm \tau ,\pm \mathbf{x}\right){\widehat{\zeta }}_{\pm }\left(\pm \tau ,\pm {\mathbf{x}}^{\prime }\right){|0_{\pm }\rangle }_{\mathrm{qdS}}\approx P_{\zeta }\left(1\pm \Delta {\mathcal{P}}_v\right)$$

(30)

where $r=|\mathbf{x}-{\mathbf{x}}^{\prime }|$ and ${\mathcal{P}}_{\zeta }$ is the near-scale-invariant part of the power spectrum $P_{\zeta }$:

$$P_{\zeta }={\displaystyle \frac{H^2}{8\pi ϵ}}\approx A_s{\left({\displaystyle \frac{k}{k_{*}}}\right)}^{n_s-1}.$$

(31)

where $n_s=1-2ϵ-\eta$. In standard inflation (SI) calculations, only the term + (i.e., the wave with $e^{-ik\tau }$ in Equation (26)) is considered, and the slow-roll corrections are usually neglected. This results in the (near) scale-invariant power spectrum (See Appendix G in [25] for more details). In DSI, the correction, $\Delta {\mathcal{P}}_v$ comes with positive and negative contributions to the (near) scalar-invariant part of Equation (31) at parity conjugate points because we apply (quantum mechanically) the time reversality equation given in Equation (28) to the quasi-de Sitter vacuums $|0_{\pm }{\rangle }_{\mathrm{qdS}}$ (Equation (23)) at the parity conjugate regions.

From the latest Planck data [51], $A_s=2.2\times {10}^{-9}$ and the value of spectral index $n_s$ corresponding to the pivot scale $k_{*}=a_{*}{H}_{*}=0.05 {\mathrm{Mpc}}^{-1}$ is

$$n_s=1-2{ϵ}_{*}-{\eta }_{*}\approx 0.9634\pm 0.0048\left(\mathrm{Planck}\mathrm{TT}+\mathrm{TE}+\mathrm{EE}\right)$$

(32)

In DQFT inflation (DSI), we include both terms as a direct-sum, which results in Equation (27) and leads to Equation (30) with

$$\begin{array}{c}\hfill \Delta {\mathcal{P}}_v=\left(1-n_s\right)Re\left[{\displaystyle \frac{2}{H_{3/2}^{\left(1\right)}\left(\frac{k}{k_{*}}\right)}}{\displaystyle \frac{\partial H_{{\nu }_s}^{\left(1\right)}\left(\frac{k}{k_{*}}\right)}{\partial {\nu }_s}}{|}_{{\nu }_s={\displaystyle \frac{3}{2}}}\right].\end{array}$$

(33)

Note that the ± in Equation (30) comes from the difference in the quantum vacuum in Equation (23) in the parity conjugate regions emerging from the time-reversal operation given in Equation (28). In other words, the additional term $\Delta {\mathcal{P}}_v$ in Equation (30) corresponds to a purely anti-symmetric contribution that changes sign at parity conjugate points. In spherical coordinates, the parity transformation maps a point at radial distance r from angular position $(\theta ,\phi )$ to its antipode, $(\pi -\theta ,\pi +\phi )$. This transformation cannot be realized by any rotation, and thus represents a discrete global symmetry distinct from statistical isotropy. It is important to emphasize that this global spatial parity asymmetry is unrelated to the observed birefringence in the CMB polarization, which arises from parity-violating interactions with well-defined transformation properties [52].

4.1. CMB Temperature Sky Fluctuations

On very large, super-horizon scales, the dominant contribution to CMB anisotropies arises from the Sachs–Wolfe effect [53], which corresponds to the gravitational redshift of the CMB photons due to small variations in curvature perturbations. This results in a one-to-one correspondence between temperature and metric fluctuations: $\Delta T\left(\widehat{n}\right)\sim \zeta$. The observed CMB temperature fluctuation map $\mathrm{T}\left(\widehat{n}\right)\equiv \Delta T\left(\widehat{n}\right)/T_0$ over the sky position $\widehat{n}$ can be decomposed into spherical harmonics:

$$\mathrm{T}\left(\widehat{n}\right)=\sum a_{\ell m}{Y}_{\ell m}\left(\widehat{n}\right),a_{\ell m}=\int sin\theta d\theta d\phi \mathrm{T}\left(\widehat{n}\right)Y_{\ell m}^{*}\left(\theta ,\phi \right);C_{\ell }=\sum _m{\displaystyle \frac{|a_{\ell m}{|}^2}{2\ell +1}}$$

(34)

For a Gaussian field, the statistical properties of the Fourier-like harmonic coefficients $a_{\ell m}$ are characterized by the angular power spectrum $C_{\ell }$, as defined above. For antipodal (parity-conjugate) sky positions $(-\widehat{n})$, the spherical harmonics $Y_{\ell m}$ satisfy the following:

$$Y_{\ell m}(-\widehat{n})=Y_{\ell m}\left(\pi -\theta ,\pi +\phi \right)={\left(-1\right)}^{\ell }{Y}_{\ell m}\left(\theta ,\phi \right)={\left(-1\right)}^{\ell }{Y}_{\ell m}\left(\widehat{n}\right).$$

(35)

The additional DSI term $\pm \Delta v$ in Equation (27) corresponds to a purely anti-symmetric contribution that changes sign at parity conjugate points (due to the time-reversal operation). This translates into an additional purely anti-symmetric temperature fluctuation $\delta T$ in the DSI model superimposed on the standard inflation (SI) contribution ${\mathrm{T}}^{SI}\left(\widehat{n}\right)$:

$$\begin{array}{cc}\hfill {\mathrm{T}}^{DSI}\left(\widehat{n}\right)& ={\mathrm{T}}^{SI}\left(\widehat{n}\right)\left(1+\delta \mathrm{T}\right)\hfill \\ \hfill \delta \mathrm{T}\left(\widehat{n}\right)& =-\delta \mathrm{T}\left(-\widehat{n}\right)\Rightarrow \delta \mathrm{T}\left(\widehat{n}\right)=\sum _{\ell ,m}{\left(-1\right)}^{\ell +1}\delta a_{\ell m}{Y}_{\ell m}\left(\theta ,\phi \right)\hfill \end{array}$$

(36)

This shows that when the fluctuations $\delta a_{\ell m}$ are smoothed and do not vary significantly from one multipole to the next, the additional anti-symmetric DSI contribution results in an alternating negative (for even multipoles) and positive (for odd multipoles) modulation on top of the SI signal.

In the Sachs–Wolfe regime, we can relate the 2D angular power spectrum $C_{\ell }$ to the metric 3D metric power spectrum $P_{\zeta }\left(k\right)$ by a simple projection (see, e.g., [54]):

$$C_{\ell }={\displaystyle \frac{2}{9\pi }}\int {\displaystyle \frac{dk}{k}}{\mathcal{P}}_{\zeta }\left(k\right)j_{\ell }^2\left(k/k_s\right)$$

(37)

where $k_s=7\times {10}^{-5}{\mathrm{Mpc}}^{-1}$ and $j_{\ell }\left(z\right)$ are spherical Bessel functions of the first kind. Because the time-reversal symmetry $\mathcal{T}$ is broken by the slow-roll parameters in Equation (28), we expect the quantum field during inflation to evolve asymmetrically in time at parity-conjugate points in physical space (resulting in Equation (27) and Equation (30)). Following Equation (30) and Equation (36), we obtain an angular power spectrum $C_{\ell }$ of the CMB that exhibits an excess of power in odd multipoles compared to even ones, as described below in Equation (38).

In Figure 3, we can visually identify this parity asymmetry by the strong resemblance between the odd-parity component of the temperature map and the original CMB maps from Planck data [55]. This reflects that the CMB temperature maps contain an additional pure odd-parity component, as indicated by Equation (36). The DSI framework naturally generates this additional odd symmetry.

It is worth noting that if we take the average of even and odd power spectra, we recover the well-known near-scale-invariant power spectrum of standard inflation (SI) with $\Delta {\mathcal{P}}_v\simeq 0$. The advantage of the DSI model is that it requires no additional free parameters: the parity asymmetry depends solely on the spectral index $n_s\approx 0.9634\pm 0.0048$ at the pivot scale $k_{*}=0.05{\mathrm{Mpc}}^{-1}$ ($\ell \approx 800$), as measured by Planck [51]. Inflationary quantum fluctuations in DSI are non-Markovian (See Section 5.4 of [25]). So, we compute here the scale-dependent features of power spectra for the large-wavelength modes that affect the low-ℓ features of the CMB angular power spectrum.

4.2. DSI vs. SI Model Comparison and Simulations

To assess the significance of the observed parity asymmetry in the CMB, we compare ${10}^6$ simulated realizations of the data under two models: the SI model and the DSI model. Specifically, we evaluate the posterior probability $p\left(M\right|D)$ of each model M given the data D.

This approach contrasts with the standard practice in the CMB community, which typically estimates the likelihood $p\left(D\right|M)$—the probability of the data given a specific model—to assess the significance of low-multipole anomalies. However, this likelihood-based method presupposes the model and can lead to inflated uncertainties, especially because the $\Lambda$CDM model with SI generally predicts more large-scale power than is observed. This mismatch increases the sampling variance, thereby reducing the apparent significance of observed parity anomalies. For example, the low measured quadrupole $C_2$ has $p\left(D\right|M)=2.62\%$, while the posterior value is 29 times smaller—$p\left(M\right|D)=0.09\%$, as shown in

Table 1. Parity posterior probability $p\left[M\right|D]$ ($\times {10}^2$, i.e., in %) of a model given the data (based on ${10}^6$ sky realizations of the data). Each line corresponds to a different CMB parity indicator. We compared two different models: Standard Inflation quantum fluctuations with LCDM (SI) and direct-sum inflationary quantum fluctuations (DSI). Data is estimated from the Planck 2018 masked SMICA component separation map. Very similar results are found for the other maps (see dashed lines in Figure 4). In the last column, ’ratio’ refers to the ratio of posterior probabilities between the DSI and SI predictions. In the second column, we also show for reference the non-posterior likelihood $p\left[D\right|M]$.

Parity Indicator	SI $\mathit{p}\left[\mathit{D}\right|\mathit{M}]$	SI $\mathit{p}\left[\mathit{M}\right|\mathit{D}]$	DSI $\mathit{p}\left[\mathit{M}\right|\mathit{D}]$	Ratio DSI/SI
$C_2$	2.62%	0.09%	3.3%	37
$R^{TT}$	1.0%	0.7%	39.5%	56
$Z^1$	3.89%	1.12%	45.3%	40
$C_2,R^{TT}$	0.12%	0.003%	1.96%	653
$Z^1,R^{TT}$		0.45%	34.6%	77
$Z^1,C_2$		0.016%	2.65%	166

.

Both the data and simulations are processed in the same way. Full details of the simulation setup are provided in Appendix D of [25]. Here, we focus on the comparison of the ${10}^6$ all-sky-masked realizations of the measured $C_{\ell }$ power spectrum. We used the $C_{\ell }$ values measured in the Planck 2018 analysis. We consider four Planck component-separated map estimates: Commander (${\mathrm{COMM}}_m$), SMICA (${\mathrm{SMIC}}_m$), SEVEM (${\mathrm{SEVE}}_m$), and NILC (${\mathrm{NILC}}_m$), each analyzed at $N_{\mathrm{side}}=2048$. The corresponding sky fractions are $f_{\mathrm{sky}}=88.8\%$, $84.2\%$, $83.8\%$, and $78.6\%$, respectively. The appropriate mask is applied to each simulation set. We use the HEALPix 3.83 software package (https://healpix.sourceforge.io/ (accessed on 2 December 2024)) for all-sky map analysis and visualizations. The SI model corresponds to the Planck 2018 best-fit, scale-invariant $\Lambda$ CDM angular power spectrum, denoted $C_{\ell }^{\mathrm{SI}}$ (shown as a black dashed line in the top panel of Figure 4) and goes through the middle of the $C_{\ell }$ measurements (cyan line and shaded region 68%), as expected. The DSI model introduces a parity-violating modulation to the SI spectrum as follows:

$$C_{\ell }^{\mathrm{DSI}}=C_{\ell }^{\mathrm{SI}}\left[1+{(-1)}^{\ell +1}\Delta {\mathcal{C}}_{\ell }\right]$$

(38)

where the fractional modulation $\Delta {\mathcal{C}}_{\ell }$ is

$$\Delta {\mathcal{C}}_{\ell }={\displaystyle \frac{1}{C_{\ell }^{\mathrm{SI}}}}{\int }_0^{k_c}{\displaystyle \frac{dk}{k}}{A}_s{\left({\displaystyle \frac{k}{k_s}}\right)}^{n_s-1}{j}_{\ell }^2\left({\displaystyle \frac{k}{k_s}}\right)\Delta {\mathcal{P}}_v\left(k\right)$$

(39)

Here, $\Delta {\mathcal{P}}_v\left(k\right)$ is given in Equation (33), and the standard $\Lambda$CDM angular power spectrum is as follows (see Equation (37)):

$$C_{\ell }^{\mathrm{SI}}={\int }_0^{\infty }{\displaystyle \frac{dk}{k}}{A}_s{\left({\displaystyle \frac{k}{k_{*}}}\right)}^{n_s-1}{j}_{\ell }^2\left({\displaystyle \frac{k}{k_s}}\right)$$

(40)

The SI prediction corresponds to $\Delta {\mathcal{P}}_v=0$, while the DSI one is given by Equation (33). A cut-scale $k_c=0.02k_{*}$ is used in Equation (39) as Equation (33) is accurate enough for low-ℓ or large angular scales. This cutoff scale is also related to the coarse-graining scale of stochastic inflation [25], which determines the large-wavelength modes that have already become classical on the onset of inflation while the mode $k_{*}$ exits the horizon. In DSI, these large-wavelength modes create parity-asymmetric inhomogeneities that correct the FLRW spacetime on large scales. The parity asymmetry ceases to exist at small scales because the $\mathcal{P}\mathcal{T}$ symmetry is recovered at short-distance scales. This can be seen in Figure 11 of [25]. An intuitive explanation for this is that large-wavelength modes experience quantum gravitational effects more than those of short-wavelength modes. For the small scales $k>k_c$, the predictions of DSI become almost the same as with the (near) scale-invariant power spectrum as contributions from $\Delta {\mathcal{P}}_v$ term become negligible.

The resulting modulation in Equation (38) oscillates with amplitude $\sim 20\%$ for multipoles $\ell <10$, producing a measurable parity asymmetry (see dashed red line in top panel of Figure 4). We generate ${10}^6$ Monte Carlo realizations for both SI and DSI models, allowing a direct and statistically robust comparison with the Planck data.

In Figure 4 and Table 1 we present how the posterior probabilities from the ${10}^6$ realizations of the observational data very significantly prefer the mean DSI model prediction over the SI model. We compare the quadrupole amplitude $C_2=C_{\ell =2}$, the even-to-odd multipole ratio of angular power spectra:

$$R^{TT}={\displaystyle \frac{{\sum }_{\ell =even}^{\ell max=20}\ell (\ell +1)C_{\ell }}{{\sum }_{\ell =odd}^{\ell max=20}\ell (\ell +1)C_{\ell }}}$$

(41)

and the mean $Z^1\equiv <Z>$ of the parity map in Figure 5.

5. Conclusions

In this paper, we underscore the pivotal role of unitary quantum field theory in curved spacetime (QFTCS) in elucidating the intricacies of early-universe physics. Contrary to prevailing notions, we contend that the issue of unitarity in QFTCS is not inherently tied to quantum gravity at Planck scales, but rather stems from throwing away the vital role of $\mathcal{P}\mathcal{T}$ transformations in the construction of vacuum for quantum fields in curved spacetime. In the context of de Sitter spacetime, which is a $\mathcal{P}\mathcal{T}$-symmetric manifold, we illustrated that one can restore unitarity of QFTCS by taking into consideration geometrical aspects of spacetime.

With the $\mathcal{P}\mathcal{T}$-based formulation of the Schrödinger equation given in Equation (1), we embrace a new understanding of quantum theory where a single quantum state is expressed as direct-sum of two components that describe the parity conjugate regions of physical space with opposite arrows of time. The DQFT framework constructs quantum fields with opposite arrows of time at parity conjugate regions of physical space. This new geometric construction of quantum fields echoes with discrete (a)symmetries of the spacetime manifold. Within this new description, the spacetime horizon acts as a $\mathcal{P}\mathcal{T}$ mirror that contains any information within the horizon in the form of pure states. This reinstates unitarity in de Sitter spacetime, which was previously thought to be lost in standard frameworks of quantization.

Figure 1 and Figure 2 visually represent this concept, illustrating how a hypothetical classical observer consistently experiences pure states without encountering information loss beyond the horizon. The DQFT framework can be applied to any curved spacetime without any necessity of $\mathcal{P}\mathcal{T}$ symmetry in the manifold. The construction of DQFT only prescribes that every quantum field should be expressed via two components describing parity conjugate regions of physical space with opposite arrows of time, where the quantum vacuum is a geometrical direct-sum based on $\mathcal{P}\mathcal{T}$ operations. With this construction, quantum fields could evolve asymmetrically at parity conjugate points if the time-reversal symmetry is broken. Thus, in manifolds like de Sitter spacetime, where the manifold is $\mathcal{P}\mathcal{T}$-symmetric, quantum fields geometrically obey the symmetry and lead to equal two-point correlations at parity conjugate regions of physical space.

Applying the framework of unitary QFTCS to inflationary quantum fluctuations, we derived a prediction for a parity asymmetry in the CMB originating from time-reversal symmetry-breaking during inflation. It is well-known that time is a parameter in quantum theory. If one considers quantum fluctuations in gravitational physics, all the time-dependent quantities that describe the dynamics of spacetime act as parameters. Thus, in quantum gravity (even at the linearized level, which is the case for inflationary quantum fluctuations), time reversibility is much more complex than a naive understanding of time coordinate reflection. In our construction, the slow-roll parameters that describe inflationary spacetime dynamics break the $\mathcal{P}\mathcal{T}$ symmetry of the de Sitter vacuum in DQFT, leading to parity-asymmetric evolution of the inflationary quantum fluctuations. We have shown that this leads to a non-trivial correlation between anti-podal points ($\left(\theta ,\phi \right)$ and $\left(\pi -\theta ,\pi +\phi \right)$) of the CMB sky. The striking correlation illustrated in Figure 3, Figure 4 and Figure 5 between the observed hot and cold mirrored structures of the CMB map at anti-podal points provides strong support for the unitary treatment of inflationary fluctuations within the DQFT framework.

Our findings indicate that superhorizon fluctuations at anti-podes manifest as $\mathcal{P}\mathcal{T}$ conjugates, a prediction derived from direct-sum QFTCS (i.e., the direct-sum inflation, DSI). This prediction is strongly supported by the data, with a posterior probability over 650 times greater than that of standard inflation, which is based on non-unitary QFTCS (see Table 1). The observed parity asymmetry also provides a unified explanation for other large-scale CMB anomalies, such as the so-called hemispherical power asymmetry [25].

To assess the significance of the low-multipole anomalies, we evaluated the posterior probability $p\left(M\right|D)$ of each model M given the data D. This Bayesian approach contrasts with the standard practice in the CMB community, which evaluates $p\left(D\right|M)$—the likelihood of the data under the SI model. For example, the low measured quadrupole $C_2$ has $p\left(D\right|M)=2.62\%$ while the posterior value is 29 times smaller: $p\left(M\right|D)=0.09\%$, as shown in Table 1. The non bayesian approach tends to inflate uncertainties due to the excess power predicted by $\Lambda$CDM at large scales, which increases sampling variance and thereby reduces the apparent significance of observed parity asymmetry.

Our study highlights the importance of QFTCS and its non-trivial role in underpinning the quantum aspects of gravity. By sticking to the foundational aspects of QFTCS, we addressed long-standing issues associated with understanding of CMB anomalies. Our grounded approach not only sheds new light on understanding quantum gravity but also allowed us to derive predictions that necessarily addressed the observations. This approach complements and poses new challenges for past investigations [56,57] that addressed CMB anomalies as evidence of yet unknown frameworks of Planck-scale quantum gravity.