Cosmic microwave background in a non-Archimedean world

Abstract

The universe begins its expansion from dimensions below the Planck scale, where the uncertainty principle reigns. If at these dimensions the Archimedean geometry is useless, the p-adic non-Archimedean world creates an environment conducive to phenomenological modeling. The standard description of the cosmological density contrast is made by the Fourier decomposition of plane waves in the field of p-adic numbers $\mathbb {Q}_p$. The coupling of the primordial perturbations is done by coupling the phases. Since the phases are the stars of this article, we represent them in the hue-saturation-brightness color space both in $\mathbb {R}$ and in $\mathbb {Q}_p$. If the primordial perturbations interact weakly with each other, with their growth they become important enough from an energetic point of view to gravitationally interact with each other. We identify the coupling using the phase gradient operator, and extract the coupled points from the spectrum. We use pseudoentropy for the quantitative analysis of the information generated by the phases.

1 Introduction

The most circulated model for the distribution of objects in the universe implies that the current large-scale structure was formed due to the gravitational interactions of the first fluctuations in the density field. The inflationist model proposes that the fluctuations are generated by the quantum oscillations of the scalar field, assumed to be completely random. The random distribution is considered to be the most generic form of initial perturbations.

To study the primordial fluctuations, we analyze the density contrast, which we extract from the cosmic microwave background (CMB) map. Its development in Fourier series implies the independent and uniformly random distribution of the phases on the interval $[0,2\pi )$. The hypothesis of the random phase, essential in the definition of Gaussian random fields from a statistical point of view, is a minimum requirement imposed on the measurements. Scientific studies based on the analysis of Fourier modes appear more and more frequently in the specialized literature. Using a toy model of one-dimensional perturbations evolving under the Zel’dovich approximation, the paper [1] developed a statistic that quantifies the information content of the distribution of phases, in terms of an effective entropy. The paper [2] showed that phase-only reconstructions exhibit a high degree of correlation with the original distributions, showing that the spatial reconstruction of cosmological density fields depends more on phase accuracy than on amplitudes. The study [3] displayed interesting relationships between phase entropy and gravitational clustering. In contrast to the Shannon entropy method of the distribution of neighboring phase differences, this method does not require a large number of phases. The paper [4] developed an approximate theory for the probability distribution and test it using a large battery of numerical simulations. The authors found a remarkable universal form for the distribution. Using a large set of N-body simulations, the paper [5] explored the behavior of phase correlations for various configurations of wave vector triangles in Fourier space at different epochs in a range of different sampling volumes.

The p-adic calculus appeared as a part of mathematical physics relatively recently [6], in an attempt to find a mathematical framework to describe the non-Archimedian geometry for string dynamics at the size of the Planck scale. There are several formalisms for describing quantum mechanics. The work [7], develops a variant of quantum mechanics in which the canonical variables are considered p-adic. To exemplify the formalism, they consider the case of the free particle and the harmonic linear oscillator. In the paper [8] a theoretical description of adelic quantum mechanics is formulated. One of the representative consequences of this theory is that the adelic harmonic oscillator softens the uncertainty relation. The paper [9], introduced for the first time the definition of the p-adic fluorescence spectrum in analogy with the ordinary one.

Several papers have been written based on the idea that both the primordial universe and black holes are manifestations of non-Archimedian geometry. The action and gravitation field equations over the p-adic number field are investigated in [10] and analogs of some solutions to the Einstein equations are presented. In the work [11], the authors considered the classical p-adic and spatial noncommutative form of a cosmological model with Friedmann-Robertson-Walker metric coupled with a self-interacting scalar field. Within the p-adic quantum cosmology the main results concerning the de Sitter model have been presented in [12]. The consequence of this formulation is the radius discreteness of the universe. In [13], the authors discussed the relevance of the classical and quantum rolling tachyons inflation in the frame of the standard, p-adic and adelic minisuperspace quantum cosmology.

One of the main objectives of mathematics is to identify compact models for describing complex systems. Chaos theory is one of the major theories of the 20th century in physics, basically describing how relatively simple systems perform extremely complex, unpredictable movements. By definition, chaos is a process that is extremely sensitive to initial conditions. Probably the best-known example of chaos is the butterfly effect, exemplified by the simple flapping of wings in Brazil that creates a hurricane in Texas. To prove Fermat’s last theorem, Wiles used p-adic numbers. Real numbers are used to describe Euclidean geometry (sphere, ellipsoid,...) while p-adic numbers analytically describe fractal geometry. In the p-adic calculus, the chaos comes from the fact that the p-adic field $\mathbb {Q}_p$ is not ordered. Fractals have become the visual identification of chaos. The authors of the work [9] concluded that the p-adic spectrum behaves chaotically, and this is due to the fact that there is an interdependence between p-adic numbers, fractals and chaos theory.

The very early stage in the evolution of the universe is related to phenomena below the Planck scale. Since distances around the Planck scale are subject to the uncertainty principle, their measurements become impossible, making Archimedean geometry useless. We say that a field is non-Archimedian if $\vert x+y\vert \le \max \{\vert x\vert ,\vert y\vert \}$, for all x and y values in the respective field. Based on this geometry, we want to analyze the primordial perturbations. For this, we introduce the definition of the cosmological density contrast, and develop it in Fourier series in the field of p-adic numbers, and study how the phase distribution is influenced by the choice made. A main stage in the elaboration of the analysis is the establishment of a quantitative method for analyzing the phases. We are interested in how the phase correlation modifies the structure of the universe both in the continuous situation and in the p-adic situation.

The paper is structured as follows: in Sect. 2, we introduce the field of p-adic numbers and present the additive character function, which plays the role of the exponential in real analysis. We represent the frequencies in the hue-saturation-brightness color space with the help of colors and highlight the difference between the real theory and the p-adic theory. In Sect. 3, we study the correlation of Fourier modes with the help of the additive character function, and we use pseudoentropy to quantitatively measure the information.

2 Perturbations in a p-adic world

In this part, we propose to discuss the cosmological perturbations in the p-world. For this, we must remember certain technical aspects of the p-adic calculation.

We denote by $\mathbb {Q}$ the field of rational numbers. If p is a given prime number, then any rational number x can be uniquely written as $x=p^{\nu }m/n$, with m, n being integers and divisible by p, and with $\nu$ integer. The field of p-adic numbers is denoted by $\mathbb {Q}_p$ and is defined as

$$\begin{aligned} \mathbb {Q}_p=\bigcup _{\gamma =-\infty }^{\infty } \left\{ x\,|\,|x|_p=p^{\gamma }\right\} , \end{aligned}$$

where the p-adic absolute value is calculated according to the rule

$$\begin{aligned} |x|_p =\left\{ \begin{array}{ll} p^{-\nu } &{}\quad \text {if}\quad x\ne 0, \\ 0 &{} \quad \text {if}\quad x= 0. \end{array} \right. \end{aligned}$$

According to Ostrowski’s theorem, the field of rational numbers can be completed with both the regular and p-adic absolute values. Completing it with the p-adic norm, the field $\mathbb {Q}_p$ is obtained.

Any p-adic number can be written as a power series, i.e.,

$$\begin{aligned} x=p^{\nu }\sum _{j=0}^{\infty }a_jp^j,\quad a_j\in \mathbb {Z},\quad 0\le a_j\le p-1, \end{aligned}$$

and its representation is unique for $a_0\ne 0$.

An important function in our analysis is the additive character of the field of p-adic numbers $\chi _p(x)$, defined

$$\begin{aligned} \chi _p(x)=e^{2\pi i \{x\}_p},\quad \textrm{where}\quad \{x\}_p=p^{\nu }\sum _{i=0}^{-\nu -1}a_ip^i,\quad \nu <0, \end{aligned}$$

where $\{\dots \}_p$ is the fractional part of the p-adic number. The additive character satisfies the property

$$\begin{aligned} \chi _p(x+y)=\chi _p(x)\chi _p(y),\quad x,y\in \mathbb {Q}_p. \end{aligned}$$

In the specialized literature on cosmology, the notation of the exponential function with $e^{i\phi }$ is common, where $\phi \in [0,2\pi )$. In this paper we prefer to use the notation $e^{2\pi i\phi }$, and of course the phase $\phi$ will live in the interval [0, 1). This allows us to make the connection with the p-adic wave function, defined in [14] as $e^{2\pi i \{\phi \}_p}$, in which the phase $\phi$ lives in the field of p-adic numbers. It is interesting to note that for $\phi \in \mathbb {R}$ we have the property

$$\begin{aligned} e^{2\pi i\{\phi \}}=e^{2\pi i\phi }, \end{aligned}$$

where $\{\dots \}$ represents the fractional part.

The process of wave interference in the real world is understood in the following way

$$\begin{aligned} \cos \phi _1+\cos \phi _2=2\cos \frac{\phi _1-\phi _2}{2}\cos \frac{\phi _1+\phi _2}{2}\equiv A_r\cos \frac{\phi _1+\phi _2}{2}, \end{aligned}$$

where the amplitude $A_r$ depends on the phase difference $\Delta \phi \equiv \phi _1-\phi _2$ as follows

$$\begin{aligned} \Delta \phi =\left\{ \begin{array}{ll} 2\pi m&{}\quad \text {strenghten},\\ 2\pi \left( m+\frac{1}{2}\right) &{}\quad \text {weakened}, \end{array} \right. \end{aligned}$$

(1)

with $m\in \mathbb {Z}$. The property (1) is also preserved for the p-adic case in which we redefine $\Delta \phi$ in the form

$$\begin{aligned} \Delta \phi \equiv \{\phi _1\}_p-\{\phi _2\}_p. \end{aligned}$$

(2)

This interference property is also responsible for the phase coupling in the map corresponding to the cosmic microwave background.

Density contrast plays an important role in the structure and evolution of the universe. It can reveal patterns and fluctuations in the CMB, which can provide insights into cosmic structures. We define p-adic cosmological density contrast $\delta (x)$ as a superposition of plane waves

$$\begin{aligned} \delta (x)=\frac{\rho (x)-\rho _0}{\rho _0}=\int _{\mathbb {Q}_p}dk\, \tilde{\delta }(k)\,\chi _p\left( kx\right) , \end{aligned}$$

(3)

where $\rho _0$ is the mean matter density, and both $\rho (x)$ and $\rho _0$ live in $\mathbb {Q}_p$, and dk represents the Haar measure. The complex amplitude $\tilde{\delta }(k)$ is defined as

$$\begin{aligned} \tilde{\delta }(k)=\vert \tilde{\delta }(k)\vert \chi _p(\phi _k), \end{aligned}$$

(4)

where the phases $\phi _k$ are defined in the interval $[0,2\pi )$. The theory can be easily extended to $\mathbb {Q}_p^3$.

Fig. 1

Hue circle in the HSB color space for $\phi _k\in \mathbb {R}$

Fig. 2

Hue circle in the HSB color space for $\phi _k\in \mathbb {Q}_2$

Fig. 3

Hue circle in the HSB color space for $\phi _k\in \mathbb {Q}_3$

To understand the phase distribution, it is constructive to use colors, because each color corresponds to a well-determined frequency. The hue-saturation-brightness (HSB) color space has as its main hues red, green and blue, separated by angles of $120^{\circ }$, and between them at half the distance are positioned complementary hues yellow, cyan and magenta. Purplish hues were added by Newton to be able to put together the two colors at the end of the spectrum, i.e., red and blue. In Fig. 1 we represented the hue circle in the space of HSB colors for $\phi _k\in \mathbb {R}$, and in Figs. 2 and 3 for $\phi _k\in \mathbb {Q}_p$, with $p=2$ and $p=3$, respectively. It can be seen that the p-adic world is much more correlated than the real world.

A good part of the scientific community considers the structure of the universe to be continuous. The p-adic world discretizes space, and if we come from a non-Archimedean world, we should ask ourselves to what extent this influences the supposedly continuous structure of the universe. Probably time-scale calculation is the most suitable for such a situation, since it is specially designed for reconciling the continuum with the discrete.

3 Phase correlation and pseudoentropy

For the real case, the primordial perturbations are represented in Fig. 4, where the phases $\phi _k$ are in the interval $[0,2\pi )$. The phases are visualized using the colors of the rainbow where $\phi _k=0$ corresponds to red and $\phi _k=2\pi$ to violet, and are evaluated using the formula $\phi _k=2\pi n/N$, with $n\in [0,N-1]$ and $N=20$. The considered window has the dimensions $x=30$ and $y=30$. A visual analysis of this figure tells us that the phases are random because there are few places where we find two identical hues next to each other. This means that the phases are weakly correlated. In Figs. 5 and 6 the same data as those in Fig. 4 were represented, but using the additive character function, for $p=2$ and $p=3$, respectively. Unlike the real case, in these two situations the phases are strongly correlated.

In Section Annex, we showed that

$$\begin{aligned} \langle \chi _p\left( \phi _k-\phi _{k'}\right) \rangle =0, \end{aligned}$$

(5)

for $k\ne k'$. At the beginning, the primordial perturbations weakly interact with each other, but with their growth, they begin to gravitationally interact by coupling the Fourier modes. Correlated points have the property that they contribute to the increase in information in the dynamic system, which leads to the decrease in entropy. We mark the correlation by removing them from the spectrum of values, that is, giving the value $\pi /2$ to the equal phases. This operation does not change the monotony of entropy.

Fig. 4

2D phase simulation for $\phi _k\in \mathbb {R}$

Fig. 5

Phase simulation for $\phi _k\in \mathbb {Q}_2$

Fig. 6

Phase simulation for $\phi _k\in \mathbb {Q}_3$

Fig. 7

Simulations of the mean phase $\phi$ for $\phi _k\in \mathbb {R}$. Red corresponds to $\pi$

Fig. 8

Simulations of the mean phase $\phi$ for $\phi _k\in \mathbb {Q}_3$. Red corresponds to 0.48

Fig. 9

Simulations of the mean phase $\phi$ for $\phi _k\in \mathbb {Q}_7$. Red corresponds to 0.08

To measure the information from the simulations in Figs. 4, 5 and 6, we use the average phase

$$\begin{aligned} \phi =\frac{\sum _k\phi _k}{N^2}, \end{aligned}$$

(6)

where N represents the number of equal intervals into which the interval $[0,2\pi )$ is divided. The average phase is also called pseudoentropy due to the fact that it shows the same monotony as entropy.

In Fig. 7, we represented the average phase for 15 simulations in which we considered $\phi _k\in \mathbb {R}$. When the phases are completely random on the interval $[0,2\pi )$, the mean phase $\phi$ must be $\pi \sim 3.14$. The simulations in Fig. 7 show that indeed the phases fulfill this condition. In Figs. 12, 8 and 9 we simulated the average phase for $\phi _k\in \mathbb {Q}_2$, $\phi _k\in \mathbb {Q}_3$ and $\phi _k\in \mathbb {Q}_7$, respectively. The fact that the mean phase $\phi$ is below $\pi$, shows that p-adic situations are much more correlated than the real situation. On the other hand, the lack of order in the set $\mathbb {Q}_p$ generates the chaotic character of these representations: in $\mathbb {Q}_2$ we have $\phi =0.23$, in $\mathbb {Q}_3$ we have $\phi =0.48$ and $\mathbb {Q}_7$ we have $\phi =0.08$.

Another chaotic indicator of the distribution of phases in $\mathbb {Q}_p$ is represented by the sensitivity of the pseudoentropy to the number of divisions N of the interval $[0,2\pi )$. For $p=2$ and $N=18$, 19 and 20, we represented the average phase in Figs. 10, 11 and 12. The average pseudoentropies do not evolve monotonically with the value of N. They oscillate, having values of 0.45 for $N=18$, 1 for $N=19$ and 0.23 for $N=20$.

Chaos is a characteristic of complex systems, which are generally unstable to external disturbances. The initial perturbations acquire chaotic characteristics when they are modeled non-Archimedean.

Fig. 10

Mean phase $\phi \in \mathbb {Q}_2$, for $N=18$. The average of the pseudoentropies is 0.45

Fig. 11

Mean phase $\phi \in \mathbb {Q}_2$, for $N=19$. The average of the pseudoentropies is 1

Fig. 12

Mean phase $\phi \in \mathbb {Q}_2$, for $N=20$. The average of the pseudoentropies is 0.23

4 Conclusions

In this paper we studied the primordial cosmic disturbances, before the universe exceeded the Planck dimension. The most suitable geometry for this situation is the non-Archimedean one. According to Ostrowski’s theorem, there are two ways to complete the rational numbers: the rational numbers or the p-adic numbers. Since the p-mean geometry is non-Archimedean, we used it in this paper to model the primordial perturbations.

In the first step to achieve the proposed objective, we introduce the cosmological density contrast function, which we develop in Fourier series in the field of p-adic numbers. The series involves plane waves accompanied by complex amplitudes that can be expressed with the help of defined phases in the interval $[0,2\pi )$. Since we are only interested in the influence of the phases on the large-scale structure of the universe, we represent them in the hue-saturation-brightness color space. In Figs. 1, 2 and 3 we compare the real case with the p-adic case for $p=2$ and $p=3$. Even from these figures you can see what we highlight further, i.e., in the p-adic situation, the phases are much more correlated. This assumes that in simulations of the cosmic microwave background we expect to find broad islands of the same hues.

For $\phi _k\in \mathbb {R}$, we represented the primordial perturbations in Fig. 4. Compared to the p-adic case represented in Figs. 5 and 6, the real case is not correlated. For the quantitative analysis of correlations, we introduced the average phase, which we called pseudoentropy due to the fact that it has the same monotony as entropy. The simulations in Fig. 7 show us that indeed the average phase value is close to $\pi$, a value corresponding to the case where the phases are completely random. The p-adic cases represented in Figs. 8, 9 and 12 show us that the pseudoentropy value is much below $\pi$, so the phases are strongly correlated even from generation, before interacting gravitationally.

Another conclusion obtained from the analysis of primordial p-adic perturbations, consists in their chaotic behavior. In Figs. 12, 8 and 9, we represented the pseudoentropies corresponding to the simulations in $\mathbb {Q}_2$, $\mathbb {Q}_3$ and $\mathbb {Q}_7$, and we showed that the average value of the pseudoentropies 0.23, 0.48 and 0.08, respectively, is not monotonous with the increase in the p-adic number. The sensitivity to the number of divisions N of the interval $[0,2\pi )$ is represented in Figs. 10, 11 and 12. For $N=18$, 19 and 20, the average pseudoentropy oscillates taking the values 0.45, 1 and 0.23, respectively.

Not only in cosmology, in general in complex systems new methods of mathematical description of physical phenomena still not understood are intensively sought. In this paper we have shown how, with the help of statistics and number theory, models can be created beyond the mainstream of cosmology, models that can explain the necessary extra correlations in the large-scale structure of the universe.

Annex

In this section we define and calculate the correlation and the power spectrum for the p-adic case. The one-dimensional case treated in this section trivially extends to several dimensions.

Definition 1

By definition, the p-adic integral is normalized

$$\begin{aligned} \int _{\vert x\vert _p\le 1}dx=1, \end{aligned}$$

(7)

where dx is the Haar measure.

Proposition 1

The additive character has the following property [15]

$$\begin{aligned} \int _{|x|_p= p^{\gamma }} \chi _p(k x)dx =\left\{ \begin{array}{ll} p^{\gamma }\left( 1-\frac{1}{p}\right) &{} \quad \text {if}\quad |k|_p\le p^{-\gamma }, \\ -p^{\gamma -1} &{}\quad \text {if}\quad |k|_p= p^{-\gamma +1}, \\ 0 &{}\quad \text {if}\quad |k|_p\ge p^{-\gamma +2}, \end{array} \right. \end{aligned}$$

(8)

with $\gamma \in \mathbb {Z}$.

Proposition 2

Dirac’s $\delta _D$ function can be defined

$$\begin{aligned} \delta _D(k)=\int _{\mathbb {Q}_p}\chi _p(k x)dx. \end{aligned}$$

Proof

The first case analyzed is the one corresponding to $k=0$, which corresponds to the situation where $\chi _p(0)=1$. We can write

$$\begin{aligned} \int _{\mathbb {Q}_p}dx&=\sum _{\gamma =-\infty }^{\infty }\int _{|x|_p=p^{\gamma }}dx=\sum _{\gamma =-\infty }^{\infty }\left( \int _{|x|_p\le p^{\gamma }}dx-\int _{|x|_p\le p^{\gamma -1}}dx\right) \nonumber \\&=\sum _{\gamma =-\infty }^{\infty }p^{\gamma }\left( 1-\frac{1}{p}\right) =\infty . \end{aligned}$$

(9)

To obtain the result we used Eq. (7).

For the case $k\ne 0$ we apply Proposition 1. Making the notation $|k|_p=p^N$, we can write that

$$\begin{aligned} \int _{\mathbb {Q}_p}\chi _p(k x)dx =\sum _{\gamma =-\infty }^{\infty }\int _{|x|_p=p^{\gamma }}\chi _p\left( kx\right) dx=\left( 1-\frac{1}{p}\right) \sum _{\gamma =-\infty }^{-N}p^{\gamma }-p^{-N}=0. \end{aligned}$$

(10)

Equation (9) together with Eq. (10), determine the final result. $\hfill\square$

Proposition 3

The Fourier transform and its inverse can be defined as follows

$$\begin{aligned} \tilde{f}(k)=\int _{\mathbb {Q}_p}\chi _p(k x)f(x)dx, \end{aligned}$$

(11)

and

$$\begin{aligned} f(x)=\int _{\mathbb {Q}_p}\chi _p(-k x)\tilde{f}(k)dk, \end{aligned}$$

(12)

where dx and dk are Haar measures.

Proof

Inserting Eq. (12) into Eq. (11) we obtain

$$\begin{aligned} \tilde{f}(k)=\int _{\mathbb {Q}_p}dk'\tilde{f}(k')\int _{\mathbb {Q}_p}dx \chi _p\left( \left( k-k'\right) x \right) =\int _{\mathbb {Q}_p}dk'\tilde{f}(k')\delta _D\left( k-k'\right) =\tilde{f}(k). \end{aligned}$$

The proof is similar for f(x). $\hfill\square$

Definition 2

We define the power spectrum $S(k')$ [9]

$$\begin{aligned} S(k')=\int _{\mathbb {Q}_p}dr\,G^{(1)}(r)\chi _p(-k' r), \end{aligned}$$

(13)

in which the Green’s function is defined

$$\begin{aligned} G^{(1)}(r)=\frac{\langle \delta ^*(x)\delta (x+r)\rangle }{\langle \delta ^*(x)\delta (x)\rangle }. \end{aligned}$$

Proposition 4

The power spectrum appended to the density contrast has the value

$$\begin{aligned} S(k')=\delta _D(k-k'). \end{aligned}$$

(14)

Proof

To calculate Green’s function, we start from the relation

$$\begin{aligned} \langle \delta ^*(x)\delta (x+r)\rangle&=\int _{\mathbb {Q}_p}dk\int _{\mathbb {Q}_p}dk'\vert \tilde{\delta }(k)\vert \vert \tilde{\delta }(k')\vert \chi (\phi _k-\phi _{k'}) \chi _p\left( kr\right) \nonumber \\&\quad \times \int _{\mathbb {Q}_p}dk\chi _p\left( \left( k-k'\right) x\right) \nonumber \\&=\int _{\mathbb {Q}_p}dk\chi _p\left( kr\right) \int _{\mathbb {Q}_p}dk'\vert \tilde{\delta }(k)\vert \vert \tilde{\delta }(k')\vert \chi (\phi _k-\phi _{k'}) \delta _D\left( k-k'\right) \nonumber \\&=\int _{\mathbb {Q}_p}dk\chi _p\left( kr\right) \vert \tilde{\delta }(k)\vert ^2, \end{aligned}$$

(15)

and for $r=0$ we get

$$\begin{aligned} \langle \delta ^*(x)\delta (x)\rangle =\int _{\mathbb {Q}_p}dk\vert \tilde{\delta }(k)\vert ^2. \end{aligned}$$

(16)

Inserting Eqs. (15) and (16) into Eq. (13), we obtain the following expression of the power spectrum

$$\begin{aligned} S(k')&=\frac{1}{\langle \delta ^*(x)\delta (x)\rangle }\int _{\mathbb {Q}_p}dk \vert \tilde{\delta }(k)\vert ^2\,\int _{\mathbb {Q}_p}dr\chi _p\left( \left( k-k'\right) r\right) \\&=\frac{1}{\langle \delta ^*(x)\delta (x)\rangle }\int _{\mathbb {Q}_p}dk \vert \tilde{\delta }(k)\vert ^2\delta _D\left( k-k'\right) . \end{aligned}$$

Treating the Dirac function as a constant, since it can have the values 0 or $\infty$, the proof is complete. $\hfill\square$

Remark 1

The power spectrum formula in Eq. (14) tells us that the power spectrum is 0 everywhere where $k\ne k'$. Since the power spectrum is directly proportional to the correlation, we can write

$$\begin{aligned} 0=\langle \delta ^*(k')\delta (k)\rangle =\langle \vert \tilde{\delta }(k')\vert \vert \tilde{\delta }(k)\vert \chi _p\left( \phi _k-\phi _{k'}\right) \rangle , \end{aligned}$$

for $k\ne k'$. Since we assumed that the amplitudes are constant and have the value 1, we obtain $\langle \chi _p\left( \phi _k-\phi _{k'}\right) \rangle =0$, for $k\ne k'$.