# Gas of Baby Universes in JT Gravity and Matrix Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generating Functional in Matrix Models

**Generating functional.**We consider ensemble of $N\times N$ Hermitian matrices [16,17,18,19] with potential $V\left(M\right)$. Let

**Particular case.**We consider a special case

**Double scaling limit.**This limit in matrix models has been introduced in [23], see [24,25,26] for review and refs therein. All the correlation functions ${\mathcal{Z}}_{n}^{matrix}({\beta}_{1},\cdots ,{\beta}_{n})$ in principle could be derived if the potential $V\left(x\right)$ or the spectral density/spectral curve $\rho \left(\mu \right)$ is known. To get a connection of the matrix model with JT gravity one has to go to the double scaling limit, see [1]. Consider a matrix model with a non-normalized spectral density

**Resolvents.**Similarly one has generating functional for correlation functions of resolvents

## 3. Generating Functional in JT Gravity

**Baby universes.**In cosmology [30,31,32,33,34,35,36,37,38], one usually deals with baby universes that branch off from, or join onto, the parent(s) Universe(s). In matrix theories one parent is a connected Riemann surface with arbitrary number of handles and at least one boundary. We assume that the lengths of boundaries of baby universities are small as compare with the boundary length of the parent, see Figure 1. Baby universes are attached to the parent by necks that have restricted lengths of geodesics at which the neck is attached to the parent, We assume that the lengths of the boundaries of baby universes are small compared to the length of the boundary of the parent, see Figure 1. Baby universes are attached to the parent with the help of thin necks. Thickness of the neck is defined as the geodesic length of the loop located at the thinness point of the neck, and this length is assumed to be essentially smaller than the length of theboundary of the parent, see Figure 1b,c. There are also restrictions on the area of the surface of baby universes, see [39,40] for more precise definitions. Cosmological baby universes in the parent-baby universe approximation interact only via coupling to the parent universes, that themselves interact via wormholes. In matrix models the baby universes always interact via their parents too and parents interact via wormholes, Figure 1b,c). One can expect that at large number of baby universes interaction between different parts of the system increases and this leads to phase transition (an analog of the the nucleation of a baby universe in [32]). We interpret the matrix partition function $\mathfrak{Z}$, defined by equation (22) as a partition function of the gas of baby universes.

## 4. Double Scaling Limit for the GUE

## 5. Deformation by an Exponential Potential

**Fine tuning.**It is obvious that fixing from the beginning the location of the eigenvalue one immediately gives restrictions on parameters of the potential of the matrix model. If we want to shift the location of the eigenvalues, ${S}_{[a,b]}\to {S}_{[0,b-a]}$ we have to make a shift in the potential, $V\left(x\right)\to V(x+a)$. For the quadratic potential this shift produces the linear term $\mathsf{\Delta}V\left(x\right)=jx$ and j can be determined from the location of the left point of the cut. For higher polynomial interaction the shift produces the linear term in the LHS of singular equation, as well as change of coupling constants. The shift in the exponential potential produces just a multiplication on positive constant.

## 6. Matrix Model for JT Gravity

#### 6.1. Potentials for Non-Normalized Density Distribution

#### 6.2. Effective Energy

#### 6.3. Phase Transition

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Gas of baby universes. (

**a**) One parent, (

**b**) two parents connected by the wormhole, (

**c**) three parents by the wormhole. Here $|{\beta}_{1}|>>|{\beta}_{i}|$ for $i\ge 2$ and ${b}_{i}\le {b}_{c}$ for $i\ge 2$.

**Figure 2.**Four different choices of perturbations of the gaussian ensemble by the exponential potential ${V}_{1}=J{e}^{\omega x}$: (

**a**) $J>0,\phantom{\rule{0.166667em}{0ex}}\omega >0$; (

**b**) $J>0,\phantom{\rule{0.166667em}{0ex}}\omega <0$; (

**c**) $J<0,\phantom{\rule{0.166667em}{0ex}}\omega >0$; (

**d**) $J<0,\phantom{\rule{0.166667em}{0ex}}\omega <0$.

**Figure 3.**Non–normalized density plot $\rho =\rho \left(\lambda \right)$ for the exponential potential with negative $\omega =-1$ (

**a**) and positive $\omega =1$ (

**b**) and different regularization parameter $\mathsf{\Lambda}$.

**Figure 4.**The plot of non–normalized density for the quadratic potential deformed by the exponential potential for $\omega =1$ and different values of the regularization parameter $\mathsf{\Lambda}$: $\mathsf{\Lambda}=5$ for (

**a**) and $\mathsf{\Lambda}=6$ for (

**b**).

**Figure 5.**(

**a**) Relations between ${m}^{2}$ and J for fixed $\mathsf{\Lambda}$ and $\omega $. (

**b**) ${J}_{cr}$ vs ${m}^{2}$ at $\omega =1$. The legend is the same as at (

**a**).

**Figure 6.**The potential supported the density ${\rho}_{norm,0}\left(E\right)$ for different parameter $\mathsf{\Lambda}$.

**Figure 9.**(

**a**) The points show the values of ${J}_{cr}$ for $\rho \left(\lambda \right)$ given by (66) for different $\mathsf{\Lambda}$. (

**b**) ${J}_{cr}$ vs ${m}^{2}$ for the same set of $\mathsf{\Lambda}$ as at (

**a**). $\mathsf{\Lambda}=5,6,7.5,10,12.5,15$ are shown by red, brown, magenta, blue, cyan and green colors. Here $\omega =-1$.

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Aref’eva, I.; Volovich, I.
Gas of Baby Universes in JT Gravity and Matrix Models. *Symmetry* **2020**, *12*, 975.
https://doi.org/10.3390/sym12060975

**AMA Style**

Aref’eva I, Volovich I.
Gas of Baby Universes in JT Gravity and Matrix Models. *Symmetry*. 2020; 12(6):975.
https://doi.org/10.3390/sym12060975

**Chicago/Turabian Style**

Aref’eva, Irina, and Igor Volovich.
2020. "Gas of Baby Universes in JT Gravity and Matrix Models" *Symmetry* 12, no. 6: 975.
https://doi.org/10.3390/sym12060975