# Baryon asymmetry of the universe in the inflationary multiverse scenario

Charu Gupta, R Srikanth

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Charu Gupta, R Srikanth. Baryon asymmetry of the universe in the inflationary multiverse scenario. 2022. hal-03912699

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# Baryon asymmetry of the universe in the inflationary multiverse scenario

Charu Gupta¹,²,* and R. Srikanth¹,†

¹Poornaprajna Institute of Scientific Research,
Bidalur post, Devanahalli, Bengaluru 562164, India

²Graduate Studies, Manipal Academy of Higher Education, Manipal - 576104, India

## Abstract

We compute the matter/antimatter asymmetry in the universe that would arise owing to the statistical dispersion in vacuum fluctuations that coherently straddle two neighboring bubble universes in the inflationary multiverse scenario, leading to entanglement between the two. The result is in agreement with the observed matter excess, for reasonable parameters of the inflationary scenario, and does not require the extension of the standard model to fulfil the Sakharov conditions. A further consequence of our model is an account of electrical charge neutrality of the universe.

## I. INTRODUCTION

The existence of baryon asymmetry in the universe, namely an excess of matter over antimatter, has long been recognized as a fundamental problem in the interface of cosmology and particle physics [1]. The matter/antimatter imbalance is usually quantified by the ratio

$$ \eta \equiv \frac{n_B - n_{\bar{B}}}{s_{\gamma}} \approx \frac{n_B}{s_{\gamma}} \approx 6.19 \pm 0.15 \times 10^{-10}, \quad (1) $$

where $n_B$ is the baryon number density and $s_γ$ the photon entropy density [2, 3]. In the context of leptonic matter, the asymmetry is expected to be of the same order, or possibly higher [4]. One aspect of this problem is the question of the electrical charge neutrality of the universe, namely that of an equal number of electrons and protons in the universe. The neutrality helps ensure that gravity remains the dominant force in the universe.

Various competing hypotheses have been adduced to explain the asymmetry, among them Refs. [5–8], by making an appeal to an interplay of cosmology and particle physics going beyond the Standard Model (SM). Typically, they require untested physics to

ensure satisfaction of the Sakharov conditions [9], viz., baryon number violation, C-symmetry and CP-symmetry violation, and thermal non-equilibrium.

In the present work, we consider baryogenesis in the scenario of an inflationary multiverse [10, 11]. We calculate the matter/antimatter imbalance in the universe that would arise from entanglement between two neighboring bubble universes. The remaining article is organized as follows. In Section II, we introduce the idea of entanglement between two neighboring bubble-universes, generated by coherent vacuum fluctuations straddling the two universes. In Section III, we show that baryon asymmetry arises through statistical dispersion in measured outcomes of the entangled state. This statistical concept of baryogenesis is discussed from an information theory perspective in Section IV. Section V discusses the relation to initial entanglement in modes of the scalar field, and the issue of electrical charge neutrality of the universe. In Section VI, we show that the model allows us to estimate $η$ at about $10^{-11}$, which is close to the observed value. Finally, Section VII presents our discussions and conclusions.

<footnote>* charu@ppisr.res.in</footnote>

<footnote>† srik@ppisr.res.in</footnote>
## II. VACUUM FLUCTUATIONS IN THE MULTIVERSE

Evidence such as the cosmic microwave background radiation (CMBR) and the relative abundance of light metals support the Big Bang origin of the universe about 13.7 Gyr ago [12]. Cosmic inflation is a purported exponential expansion of universe in the early phase of the Big Bang, which flattens and thermalizes the universe [10, 13]. The early universe at this stage is assumed to be described by the classical Einstein field equation, $R_{\mu\nu} - \frac{1}{2}R = kT_{\mu\nu} + g_{\mu\nu}\Lambda$, where $k \equiv 8\pi G/c^4$, $R_{\mu\nu}$ and $R$ are the Ricci tensor and scalar, respectively, while $T_{\mu\nu}$ and $\Lambda$ are the matter energy momentum tensor and cosmological constant, respectively.

An intrinsic energy density $\rho_{vac}$ of the vacuum is equivalent to the cosmological constant, and one commonly makes the association $\Lambda = k\rho_{vac}$, which is proportional to the inflaton potential $V(\varphi)$. In a matterless universe, the cosmological constant provides an exponential expansion, which forms the geometric basis of inflation. A 2nd order phase transition during which a scalar field $\varphi$ initially in a positive-energy false vacuum transitions to the true vacuum is believed to have driven the inflation. In this scenario, our universe begins as a bubble, corresponding to a pocket of advancing transition. The phase transition is expected to nucleate multiple such bubbles, each of which could be an embryonic universe potentially with its own set of physical laws.

In this scenario, our universe could be part of a pair of universes that are coherently nucleated [14]. Entanglement may be generated between the two embryonic universes because of quantum vacuum fluctuations that straddle them. Subsequently, the two universes could have separated off each other by virtue of the exponential expansion of space intervening between them.

Previously, Maldacena and Pimentel [15] considered entanglement between two

causally disconnected regions in a de Sitter space. Kanno [16] studied the scalar field spectrum due to entanglement in the field modes between a pair of de Sitter universes. Here, our focus will be on the implications of inter-universe entanglement for matter creation, following the phase transition of the scalar field.

The superluminal expansion during inflation creates a Hubble sphere – a region enclosed by an event horizon – around each point of the inflating universe. Vacuum fluctuations in the vicinity of the horizon potentially enable particle creation à la Hawking radiation [17, 18]. However, by itself this mechanism will produce *exactly* the same number of matter and anti-matter particles in the universe, leading to their eventual mutual annihilation. As discussed below, this difficulty can be eliminated if the vacuum fluctuations coherently straddle two neighboring universes in the multiverse.

Vacuum fluctuations are governed by the Heisenberg principle for “off the mass shell” (i.e., virtual) particles, given by:

$$ \Delta x \leq \frac{\hbar c}{\Delta E}, \qquad (2) $$

where $E$ is the relevant energy of the field quanta and $\Delta x$ is the uncertainty. The particle-antiparticle pair created by a vacuum fluctuation will be symmetric with respect to either universe: therefore it will lead to a coherent superposition of a particle in one universe and antiparticle in the other plus vice-versa. This will in general generate entanglement between the two universes.

This somewhat exotic situation of two entangled universes can arise if they are jointly nucleated in the multiverse scalar field. In the brief embryonic epoch that the two universes are largely co-spatial, quantum field modes may be coherently propagated across the two, and accordingly vacuum fluctuations can straddle the universes. As the bubble universes expand and mutually separate, the entanglement between the two universes may persist until eventually “measured”.
### III. ENTANGLEMENT OF VIRTUAL PARTICLES

We consider a vacuum fluctuation straddling the two universes that leads to the production of the pair consisting of a particle X and the corresponding antiparticle $\overline{X}$. Over N such instances of fluctuation, the joint state of the two universes is given by the maximally entangled state:

$$|\Psi\rangle = \frac{1}{2^{N/2}} (|X\rangle_1 |\overline{X}\rangle_2 + |\overline{X}\rangle_1 |X\rangle_2)^{\otimes N}. \quad (3)$$

The entanglement remains when the two universes separate into causally disconnected regions and move away exponentially fast from each other. At this point, each universe has N entities, but the specific identity of each entity (as X or $\overline{X}$) remains indeterminate. The reduced state of any one of the two universes (say, #1) is given by

$$\rho_1 = \left[ \frac{1}{2} (|X\rangle_1 \langle X| + |\overline{X}\rangle_1 \langle \overline{X}|) \right]^{\otimes N}. \quad (4)$$

With the creation of particles, new degrees of freedom are added to the universe, making it macroscopic in terms of its content. This leads each universe to “decohere” and become classical, meaning that the particles X and $\overline{X}$ assume a definite identity as X or $\overline{X}$.

Since the fluctuations occur during the trans-Planckian epoch, the mass of these particles could be as high as $m_{Pl}$, or perhaps even multiples thereof, for example if we admit scenarios as discussed in Ref. [19]. In Section V, we will argue that the particles X and $\overline{X}$ are probably massive, neutral baryons.

Consequently, the universe that we experience in practice is not an enormously entangled state as in Eq. (3), but a specific, probabilistic branch thereof. We may equivalently say that the state in Eq. (3) “collapses”, or that the mixture Eq. (4) is sampled. This sampling can be modelled as N Bernoulli trials, i.e., a sequence of N random variables

$Y_1, Y_2, Y_3, \dots, Y_N$ drawn from an i.i.d probability distribution corresponding to a fair coin toss. Here $Y_j \in \{0, 1\}$, with 0 (say) corresponding to a particle and 1 to an antiparticle being found in the given universe. By the law of large numbers, the sampling results with high probability in about $\frac{N}{2}$ particles and about $\frac{N}{2}$ antiparticles in our universe, and similarly in the partner universe, as quantified by the Chernoff bound:

$$\mathrm{Pr} [|\bar{Y} - \mu| \ge \delta\mu] \le 2e^{-N\mu\delta^2/3}, \quad (5)$$

where $\delta \ge 0$, $\bar{Y} \equiv \frac{1}{N} \sum_j Y_j$ is the sample mean, and $\mu \equiv \frac{1}{2}$ is the expectation value.

Even so, counterintuitively, the probability $\mathrm{Pr}[\bar{Y} = \mu]$ that the empirical mean is exactly the expectation value also approaches 0 quadratically fast, on account of statistical dispersion. This is reflected in the fact that the standard deviation of the sample sum $\sum_j Y_j$ is given by $\frac{\sqrt{N}}{2}$, implying a quadratic increase with N. As another way to appreciate this subtlety, we note that for the present case of a Bernoulli experiment, the probability to get equal heads and tails in N i.i.d trials, by virtue of Stirling’s approximation $N! = e^{-N} N^N \sqrt{2\pi N} [1 + O(1/N)]$, is given by

$$\binom{N}{N/2} 2^{-N} = \sqrt{\frac{2}{\pi N}} + O\left(\frac{1}{N}\right), \quad (6)$$

showing that it becomes quadratically less likely as N increases, a “surprising twist” in the manifestation of the law of large numbers (though not so surprising in retrospect!).

This implies that the probability that exactly the same number of particles and antiparticles are produced, falls quadratically fast. That is, there may be an excess of about $O(\sqrt{N})$ particles in our universe, and correspondingly an equal excess of antiparticles in our partner universe (or, vice versa). Because the two universes are causally disconnected after their embryonic phase, the $O(\sqrt{N})$ unpaired particles or anti-particles in either universe become real when the time limit imposed by the uncertainty relation Eq. (2) elapses.
As for the paired particle/anti-particles, which constitute the vast majority ($N - \sqrt{N} \approx N$) of generated matter, eventually they will mutually annihilate. However, the interplay of particle dynamics and intra-universe (i.e., within the universe) relativistic geometry will determine whether they do so as real or virtual particles. In turn, this will be vital to the energy constitution of the present universe.

Specifically, in the context of scattering of waves and particles, it is a general principle that the scattering cross-section varies inversely with wavelength. Analogously, we expect that among the paired particles, those that are sufficiently energetic will have a large interaction cross-section. As a result, aided by their given number density, they can overcome the dilution effect due to the exponentially expanding space, to rapidly recombine with anti-matter counterparts and mutually annihilate while still virtual (i.e., off-the-mass-shell). Thus, there is no energy cost to the scalar field.

By contrast, if the particle pairs are of sufficiently low energy, then correspondingly they have a lower cross-section. As a result, they are unable to recombine with their anti-matter partners while the exponential expansion of space due to inflation lasts. These (anti-)particles may typically have an uncertainty time window $\Delta t$ of the order of $t_{Pl}$, whereas inflation is believed to last several orders more, e.g., $70 t_{Pl}$ [20]. Therefore, these (anti-)particles are eventually promoted in status from virtual to real, and subsequent to the inflationary epoch their annihilation with an anti-particle counterpart contributes to reheating the universe. The energy for the reheating as well as for the created rest-mass is ultimately provided for by the driver of inflation, namely the scalar field.

For purpose of this work, we adopt a simple approach whereby (anti-)particles with wave length greater (resp., lesser) than the Hubble sphere radius are taken to be (anti-)particles of sufficiently low (resp., high) energy,

and thus destined to annihilate as real (resp., virtual) entities.

For example, if about $N = 10^{160}$ such straddling events take place, then about $\sqrt{N} \approx 10^{80}$ real particles of matter are created in our universe and as many anti-matter particles in the dual universe (or vice versa), which now becomes the anti-matter partner to our own. Furthermore, denoting by $f$ the fraction of the paired particles that turn real, the ratio

$$ \frac{\sqrt{N}}{Nf} = \frac{1}{\sqrt{N}f} \approx 10^{-80} f^{-1} \quad (7) $$

provides an estimate for $\eta$ in Eq. (1). Actually, this turns out to be an upper bound, since we haven't included the contribution due to non-straddling (i.e., intra-universe) vacuum fluctuations of the given universe. In Section VI, we present numerical estimates according to this scenario.

The law of large numbers (as embodied by the Chernoff bound) or Shannon's noiseless channel coding theorem imply that the ideal probability distribution or entropy is approached asymptotically for sufficiently large $n$. At first blush, the imbalance underlying baryogenesis seems to contradict this expected asymptotic behavior. That this is not the case can be understood generally from an information theoretic perspective as discussed in the following section.

## IV. INFORMATION THEORETIC CONSIDERATIONS

Suppose random variable $Y$, taking specific values $y$, is drawn from alphabet $\mathcal{Y}$. Denote the empirical sequence of values over $n$ i.i.d trials by $y^n$, and the associated empirical frequency vector by $\mathbf{F}(y^n)$: this is a $|\mathcal{Y}|$-dimensional vector listing the number of times symbol $y$ appears in $y^n$. We define type distance as

$$ D(y^n) \equiv \frac{1}{2} \| \mathbf{F}(y^n) - n\mathbf{P}(Y) \|, \quad (8) $$
where $\| \cdot \|$ denotes Hamming distance and $\mathbf{P}(Y)$ is the vector of the theoretical probabilities of the different symbols $y$. $D(y^n)$ is a measure of dispersion of the empirical sequence from an ideal one. For example, in a fair coin toss over the alphabet $\{H, T\}$, an empirical sequence $HHTTHH$ has a type distance of $\frac{1}{2} \left\| \begin{pmatrix} 4 \\ 2 \end{pmatrix} - \begin{pmatrix} 3 \\ 3 \end{pmatrix} \right\| = 1$. Type distance induces a concept of typicality that is closer to the concept of strong typicality than to that of weak typicality.

We define the *ultratypical* set by $U_k^{(n)} \equiv \{y^n \mid D(y^n) \le k\}$. A key property of the ultra-typical set is it is a strict subset of the *typical* set. Recollect that a typical set $A_{\varepsilon}^{(n)}$ is defined as the set of sequences $y^n$ such that $n(H(Y) - \varepsilon) \le -\log p(y_1, y_2, \dots, y_n) \le n(H(Y) + \varepsilon)$, where $H(Y)$ is the Shannon entropy associated with the probability vector $\mathbf{P}$. The definition of typicality is broad enough to allow the deviation of a randomly picked string $y^n$ from the ideal $nH(Y)$ by an amount (namely, $n\varepsilon$) which increases linearly with $n$. Ultratypicality imposes a stricter condition in that it allows at most a fixed deviation $k$ from the ideal value, independent of $n$.

Specifically, a typical set $A_{\varepsilon}^{(n)}$ has the property that given small $\varepsilon > 0$, there exists $n_0$ such that for any $n > n_0$, a randomly chosen string $y^n$ satisfies $\mathrm{Pr}[y^n \in A_{\varepsilon}^{(n)}] > 1 - \varepsilon$. By contrast, we have the following result for the ultratypical set, which highlights the statistical surprise that underlies baryogenesis in our model.

**Theorem 1.** For any fixed type-distance $k \ge 0$ and a small $\varepsilon > 0$, one can choose $n_0$ such that for $n \ge n_0$ we have $\mathrm{Pr}[y^n \in U_k^{(n)}] \le \varepsilon$.

*Proof.* Suppose i.i.d random variables $Y_1, Y_2, \dots, Y_n$, each characterized by mean $\mu$ and standard deviation $\sigma$, produce sequences $y^n$. Define $S \equiv \sum_{j=1}^n Y_j$, the sum random variable taking values $s$. By the Central Limit Theorem, the limit of the distribution random variable $T \equiv \lim_{n \to \infty} \frac{S-n\mu}{\sqrt{n\sigma}}$ is a

standard normal (Gaussian) distribution. Accordingly, in this limit the distribution $\nu(s) = \frac{1}{\sigma\sqrt{2\pi n}} \exp\left(-\frac{(s-n\mu)^2}{n\sigma^2}\right)$ characterizes $S$. For a given distance $k$, there are multiple possible sequences $y^n$, with varying sums $s$. Denote the maximum of this by $K \equiv \max_{y^n \in U_k^{(n)}} s[y^n]$. It can be shown that $K \le k\Delta_y$, where $\Delta_y$ is the difference between the maximum and minimum values of the symbols in the alphabet $\mathcal{Y}$. Importantly, $K$ is independent of $n$, as is $k$ by assumption.

As $n \to \infty$, the function $\nu(s)$ can be approximated by a constant value, namely $\nu(n\mu)$, throughout the range $[n\mu-K, n\mu+K]$. Then, $\lim_{n \to \infty} \mathrm{Pr}[y^n \in U_k^{(n)}]$ is given by

$$q_{k,n} \equiv \int_{n\mu-K(k)}^{n\mu+K(k)} \nu(s)ds \lesssim \frac{K}{\sqrt{2\pi n}}. \quad (9)$$

Therefore, given $k \ge 0$, we can always choose sufficiently large $n_0$ so that in Eq. (9) $q_{k,n} \le \varepsilon$ for all $n \ge n_0$. Specifically, $n_0 = \frac{K^2}{2\pi\varepsilon^2}$. $\square$

In the cosmological context, Theorem 1 implies that any fixed neighborhood of distance $k$ from perfect particle/antiparticle balance, no matter how large the distance, will be arbitrarily unlikely for sufficiently large $n$. Specifically, the case of an equal number of particles and antiparticles corresponds to $D(y^n) = 0$, i.e., the ultratypical set $U_0^{(n)}$. Using Eq. (9) with $\sigma = \frac{1}{2}$, we can bound the probability for this by $\mathrm{Pr}[y^n \in U_0^{(n)}] \approx \int_{\frac{n-1}{2}}^{\frac{n+1}{2}} \sqrt{\frac{2}{\pi n}} \exp\left(-\frac{4(s-n/2)^2}{n}\right) \approx \sqrt{\frac{2}{\pi n}}$, consistent with our earlier result in Eq. (6) showing that the probability for perfect baryon symmetry vanishes quadratically fast with $n$. Here the fraction $\frac{1}{2}$ in the integral limits is the continuity correction factor.

# V. BARYOGENESIS VIA NEUTRAL MASSIVE PARTICLES

There can be more than one particle species $\{X, \bar{X}\}$ that are involved in matter creation, but they must be massive, neutral
baryons for the reasons discussed below. The particles $X$ and $\bar{X}$ are presumably unstable and rapidly decay into relativistic SM quarks and leptons. As the universe cools to exit the trans-Planckian epoch, the initial imbalance introduced between particles $X$ and $\bar{X}$ is communicated to the SM matter in the hadronic epoch, where it appears as baryon asymmetry.

As an example of this idea, suppose $X$ is a massive, neutral baryon that decays according to the channel (ignoring possible mesonic intermediate decay stages)

$$X \rightarrow 2u + d + e^{-} + \bar{\nu} \quad (10)$$

in agreement with baryon and lepton number conservation. The state Eq. (4) yields about $\frac{N}{2}$ baryons $X$ and about $\frac{N}{2}$ antibaryons $\bar{X}$, with an excess of about $\sqrt{N}$ particles $X$ (or $\bar{X}$). As discussed in Section III, paired particle-antiparticles mutually annihilate either as virtual entities or as real ones, leading to reheating of the universe, whilst the unpaired excess baryons (or anti-baryons) survive to become the basis for created matter. In view of Eq. (10), when the universe cools beyond the hadron epoch, this leads to an excess of $O(\sqrt{N})$ protons over anti-protons, and, subsequently, an equal number of excess electrons over positrons surviving after the lepton epoch.

The particles $X$ in Eq. (3) must be electrically neutral. If not, then assuming charge conservation, the particle excess that leads to matter creation will also lead an electrically charged universe, contrary to observation. The neutrality of $X$ may be justified on grounds of a charge superselection principle in nature that governs vacuum fluctuations in the vicinity of event horizons. In general form, the principle may entail that Hawking radiation [17] must only produce the emission of neutral particles, as in [21], and that the Hawking emission of charged particles is forbidden (though other authors have studied this possibility).

If the particle $X$ were a massive, neutral

lepton, then the decay of $X$ would produce baryons and anti-baryons in equal number, assuming baryon number conservation, and so would the decay of $\bar{X}$, assuming C and CP symmetries. Thus no baryon asymmetry could result from an excess of $X$'s or $\bar{X}$'s. For the same reason, $X$ or $\bar{X}$ could not be a massive boson. Therefore it may be reasonable to conclude that $X$ is a baryon in the scope of this model.

A simple if important consequence of this scenario for matter creation is that there is an equal number of protons and electrons. Furthermore, whereas the protons in the universe are a product of baryon asymmetry, no corresponding lepton asymmetry is needed for the electrons. Of course, lepton asymmetry may arise from some other process, a possibility here being for the particles $X$ to be massive neutral leptons, e.g., $X \rightarrow 2u+2\bar{d}+d+\bar{u}+e^{-}+\bar{\nu} \rightarrow p+\bar{n}+e^{-}+\bar{\nu} \rightarrow p+\bar{p}+e^{+}+e^{-}+\bar{\nu} \rightarrow \bar{\nu}+\gamma$. A process such as this can lead to lepton asymmetry in the neutrino sector.

## VI. INFLATIONARY MATTER CREATION AND ENERGY CONSIDERATIONS

During inflation, coherent vacuum fluctuations straddling the two bubble universes evolves into entanglement between the two universes in the scalar field modes. In turn, because of strong coupling between the scalar field $\phi$ and SM matter field, this scalar field entanglement may become the basis of vacuum fluctuations leading to the pair production of massive (anti-)baryons such as $\bar{X}$ or $X$. In the process, the scalar field entanglement may be transformed into matter entanglement in some conserved variable, such as baryon number. We discuss a simple model of this situation.

Kanno [16] studied the effect that entanglement between the scalar field modes propagating along a pair of causally disconnected de Sitter universes, would have on their fluc-
tuation spectrum. There the entanglement is assumed to be generated by two-mode squeezing. Here we will find it convenient to model the entangled state as that produced by the action of a beam splitter on a N-quanta number state. In this case, the entangled state is given by

$$ \begin{aligned} |\Psi\rangle_{\varphi} &= (a^{\dagger} + b^{\dagger})^{N} |\text{vac}\rangle \\ &= \sum_{j=0}^{N} \binom{N}{j}^{1/2} |j, N-j\rangle_{a,b}^{\varphi}, \end{aligned} \quad (11) $$

where a and b correspond to the scalar modes propagating along the respective universe. Under the phase transition of $\varphi$ from the false vacuum to the true vacuum, the quanta of the scalar field are assumed to be transformed into the corresponding number states of the quantum fields of X and $\overline{X}$, namely:

$$ |j, N-j\rangle_{a,b}^{\varphi} \rightarrow |j, N-j\rangle_{a,b}^{X\overline{X}}. \quad (12) $$

The state $|j\rangle_x^{X\overline{X}}$ in Eq. (12) is interpreted as the state of j baryons X and $N-j$ antibaryons $\overline{X}$ in a given universe $x$. The resulting state can be suitably rewritten as the one in Eq. (3). We note that baryon number is conserved in the process, since the state $|\Psi\rangle$ in Eq. (3) has an expectation value of zero for baryon number.

Eq. (12) provides a mechanism through which the positive-energy of the false vacuum (identified with the cosmological constant $\Lambda$) is converted into matter in the form of massive neutral particles X. A simple quantitative treatment of this process would be to require

$$ \frac{d}{dt} \left( \frac{c^2}{8\pi G} \Lambda(t) + \rho_m(t) \right) \equiv \frac{d\rho_{\text{tot}}}{dt} = 0, \quad (13) $$

where $\rho_m$ denotes the mass density of the created matter. It then follows, in conjunction with the first Friedman-Einstein equation, namely

$$ \left(\frac{\dot{R}(t)}{R(t)}\right)^2 = \frac{8\pi G}{3}\rho_m(t) + \frac{c^2}{3}\Lambda(t) \quad (14) $$

that the Hubble parameter $H(t) \equiv \dot{R}(t)/R(t)$ is constant in time, and the scale factor grows exponentially fast, given by $R(t) = R_0 \exp\left\{\sqrt{\frac{\Lambda_0 c^2}{3}} t\right\}$, where $R_0$ is the initial finite size of the universe coming out of the trans-Planckian era ($t < t_{Pl}$). Despite the diminution of the cosmological constant, the universe's expansion remains uniformly exponential throughout inflation. Assuming inflation to begin at or just after the trans-Planckian era, we estimate $\Lambda_0 = (l_{Pl})^{-2} = c^3/\hbar G \approx 10^{70} \text{ m}^{-2}$.

Among the $O(N)$ paired particles produced because of the straddling fluctuations, let $f$ denote the fraction of those of sufficiently low energy (in the infrared tail of the trans-Planckian black body spectrum), for which the wavelength exceeds the Hubble distance, i.e.,

$$ \frac{\hbar c}{E} \geq \frac{c}{H} = \sqrt{\frac{3}{\Lambda_0}} = \sqrt{3} l_{Pl}. \quad (15) $$

As discussed in Section III, for these (antiparticles), the interaction cross-section is too low to counter the effect of superluminal separation due to the exponential inflation of the universe. Therefore, these particles are typically prohibited by relativistic causality from meeting their antiparticle counterparts within the time window of Eq. (2) allotted by the uncertainty principle. Hence, they lose their virtual status and become real, with the energy cost for their creation being met by the driver of inflation, namely the scalar field.

By contrast, particle pairs that are sufficiently energetic, and thus violate inequality Eq. (15), have interaction cross-section high enough to counter the effect of exponential expansion of space, allowing them to reunite with their anti-particle counterparts while still virtual. They mutually annihilate as per the requirement of the uncertainty principle, and there is no energy cost to the vacuum energy of the scalar field.

We shall now compute $f$ in Eq. (7). To this end, in the context of the par-
ticle/antiparticle pairs that do satisfy the bound Eq. (15), suppose M such pairs are produced out of the N straddling fluctuation events, i.e.,

$$Nf = M. \qquad (16)$$

After the end of inflation, these M pairs annihilate, producing as many photons and leading to the universe's reheating. We know that since the temperature of the universe falls as $T \sim R^{-1}$, and photonic entropy goes as $s_{\gamma} \sim T^3$, the total number of photons $S_{\gamma} \sim s_{\gamma}(t)R(t)^3$ is constant over the universe's evolution. Thus we assume that these M photons directly constitute the principal component of the photonic entropy budget of the universe.

By definition of M and $\eta$, the number of baryons created during baryogenesis is $\eta M$, as discussed in Section III, and thus $\eta M \approx \sqrt{N}$. Here $\eta$ is to be estimated according to our model. Then, the number of straddling fluctuation events should be

$$N \approx (\eta M)^2, \qquad (17)$$

The created particles of type X each has average energy $E_{Pl}$. As they are relativistic, we can estimate the universe's temperature $T_{\star}$ according to $(\eta M)^2 E_{Pl}/V = a(T_{\star})^4$. We thus have

$$T_{\star} = \left( \frac{E_{Pl} \eta^2 M^2}{aV} \right)^{1/4} = (\eta^2 M)^{1/4} T_{Pl}, \quad (18)$$

which is much larger than Planck temperature if

$$\eta^2 M \gg 1. \qquad (19)$$

Then, the fraction $f$ of particle pairs that become real, and eventually give rise to reheating, will be a very small fraction of the “infrared” tail of the distribution.

Letting $B(\nu, T) \equiv \frac{2h\nu^3}{c} \frac{1}{e^{h\nu/k_B T} - 1}$ be the Planck distribution, in the limit $h\nu \ll k_B T_{\star}$, we estimate this fraction to be

$$f = \frac{1}{\sigma T_{\star}^4} \int_{0}^{\nu_{Pl}/\sqrt{3}} B(\nu, T_{\star}) d\nu \\ \approx \frac{2k_B}{c\sigma T_{\star}^3} \int_{0}^{\nu_{Pl}/\sqrt{3}} \nu^2 d\nu \approx 10^{-69}. \qquad (20)$$

Inserting Eq. (17) in Eq. (16), we have

$$\eta^2 M f = 1. \qquad (21)$$

Noting that by the current estimate of the number of baryons in the universe, $\eta M = 10^{80}$, we estimate from Eqs. (20) and (21) that $\eta = 10^{-11}$, which is close to the observed value of $10^{-10}$.

Thus far, we ignored the non-straddling (i.e., intra-universe) fluctuations that are likely at least $N$. Let $g (\ge 2)$ be the ratio of the total to straddling fluctuations. In place of Eq. (16), we will have $M = gNf$. Furthermore, $T_{\star} \rightarrow T_{\star}g^{1/4}$ and correspondingly $f \rightarrow f/g^{3/4}$. Accordingly, $\eta \rightarrow \eta/g^{1/4}$. Even for rather large $g$ (say, 10), we find $g^{1/4} = O(1)$. At most, this slightly lowers our above estimate of $\eta$.

## VII. CONCLUSION AND DISCUSSIONS

Here we calculated the matter/antimatter asymmetry in the universe that would arise owing to vacuum fluctuations that coherently straddle two neighboring bubble universes, leading to entanglement between the two. Such a phenomenon is possible when the two universes are nucleated together in the inflationary multiverse scenario. The resulting estimate of $\eta \approx 10^{-11}$ is in agreement with the observed matter excess, for reasonable parameters of the inflationary scenario. Our model does not require the extension of the standard model to fulfil the Sakharov conditions. A further consequence of our model is an account of the electrical charge neutrality of the universe.

Generalizing the above scenario for matter creation, we may consider the possibility of baryon asymmetry arising from entanglement between multiple universes. In this case, the universes are assumed to be coherently nucleated during inflation, leading to multi-particle entanglement. As a particle and its antiparticle form a pair,
it seems that such entanglement should involve an even number of universes. A simple generalization of Eq. (3) for the case of four universes would be $|\Phi\rangle = \left(\begin{matrix} 4 \\ 2 \end{matrix}\right)^{-1/2} (|XX\overline{X}\overline{X}\rangle + |X\overline{X}\overline{X}\overline{X}\rangle + |X\overline{X}\overline{X}\overline{X}\rangle + |\overline{X}\overline{X}\overline{X}\overline{X}\rangle + |\overline{X}\overline{X}\overline{X}\rangle + |\overline{X}\overline{X}\overline{X}\overline{X}\rangle)$, where for brevity we use the notation whereby $|\overline{X}\overline{X}\overline{X}\overline{X}\rangle \equiv |\overline{X}\rangle_1 |X\rangle_2 |X\rangle_3 |\overline{X}\rangle_4$. Over a number of such processes, each of the $m$ universes will feature an overall matter or anti-matter content, but the total baryon number of the $m$ universes will be zero.

The statistical argument underlying our approach can also be employed in a stand-alone universe (as against a dual- or multi-universe) scenario to create matter by only partially satisfying the Sakharov conditions. Suppose there was a specific reaction $B \rightarrow C_1+C_2+C_3$ in the early universe, where $B$ is a baryon and the $C_j$'s are not. Under assumption of non-equilibrium, the reverse reaction is relatively suppressed, leading to a nett baryon number non-conservation. Now, C

and CP symmetry will ensure that the counterpart anti-baryon reactions $\overline{B} \rightarrow \overline{C}_1 + \overline{C}_2 + \overline{C}_3$ are equally probable. During the window of time where these non-standard reactions are allowed, either of these baryon-number violating reactions may happen as random i.i.d processes. Suppose $N$ such reactions occur. Analogous to our above argument, we conclude that at the end of the run, there will be about $\sqrt{N}$ excess of baryons $B$ or anti-baryons $\overline{B}$. The key observation is that even if $C$ and $CP$ symmetries hold, statistical fluctuations due to baryon number non-conservation may account for the observed baryon asymmetry. This idea may merit further study.

## ACKNOWLEDGMENTS

The authors thank C. Sivaram, Kenath Arun and Khushboo Dixit for useful discussions. CG acknowledges financial support of the Admar Mutt Education Foundation.

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