The Linear Dilaton in Cosmology and Particle Physics

1. Introduction

Theories with a warped extra dimension were proposed by Lisa Randall and Raman Sundrum [1] as a solution of the hierarchy problem. These theories contain an anti de Sitter (AdS) background metric and two branes: an ultra-violet (UV) brane, and an infra-red (IR) brane where the Higgs is either located, or at least toward which its profile is localized if the Higgs is propagating in the bulk. Moreover, the theory contains a massless radion, while the brane distance is undefined as the radion potential is flat. To make the radion massive, the conformal invariance of the theory needs to be explicitly broken by a bulk potential $V\left(\varphi \right)$ and brane potentials ${\lambda }_a\left(\varphi \right)$, in terms of a stabilizing bulk field $\varphi$, which then gives a mass to the radion and introduces and extra excitation around the interbrane separation [2]. Moreover, the same authors proposed a theory with an infinite extra warped dimension and a single brane where the Standard Model (SM) sector is localized [3]. This model does not solve the hierarchy problem (one would require e.g., supersymmetry in the brane to achieve this task) but gave rise to braneworld cosmologies where the location of the brane is subject to the Friedmann equations and plays the role of the scale factor of the universe. An interesting property of these theories, with AdS geometry, is that they have an equivalent dual defined in the boundary of the AdS space, the holographic dual, known to be a non-perturbative conformal field theory (CFT) [4].

Interestingly enough there is another warped geometry for which the holographic dual is known: the linear dilaton (LD) background metric for which the holographic dual is Little String Theory (LST), a six-dimensional theory compactified on a torus [5,6]. The particle physics phenomenology of LD metrics was done in Refs. [7,8], mainly with the goal of lowering the string scale to the TeV to find experimental signatures at present and future colliders. In the case considered in Refs. [7,8] the warped geometry does not solve the hierarchy problem between the TeV and the Planck scales, which is left to be solved by string theory. In this review we will show that there are other possibilities of the LD metric for particle physics for which the string scale is some orders of magnitude below the Planck scale, while the low energy scale is at the TeV, say midway between the original LD proposal of Ref. [7,8] and the AdS theory [1]. Moreover we have explored braneworld cosmologies based on the LD background, which provides an interesting possibility for the holographic fluid to describe cosmological dark matter.

The contents and summary of this review are as follows. In Section 2 we will present the general structure of the dilaton-gravity model in the presence of a general class of metrics which contain as particular cases the AdS and LD ones. The background vacua in the presence and absence of a 5D black hole (BH) are also computed. Applications to particle physics are presented in Section 3 were we show that there is a solution for which $M_5\sim {10}^{-4}{M}_4$, where $M_5$ and $M_4$ are the Planck scales in five and four dimensions, respectively. The warp factor is solving the hierarchy between the low energy scale $\eta \sim$ TeV scale and $M_5$, while the string theory has to be responsible for fixing the hierarchy between $M_5$ and $M_4$. It is also proven that there is a singularity in proper coordinates at $y=y_s=1/\eta$ and, if the interval is extended up to the singularity $(y\in [0,y_s])$ the bulk propagator is a gapped continuum for $\sqrt{s}>m_g=3\eta /2$. Applications to braneworld cosmology are done in Section 4 for the general class of models presented in Section 2. In the presence of the BH there is an unconventional cosmology with a holographic fluid which red-shifts with the universe expansion in different ways. In particular for the case of AdS the holographic fluid is dark radiation while for LD it is pressureless matter. The properties of the holographic fluid for the case of the LD background are studied in Section 5 where it is shown that it can be a constituent (or the whole) of the cosmological dark matter (DM) of the universe. As the holographic fluid is feebly coupled with matter, under the assumption that the cosmological inflaton is only coupled to the SM, the fluid density will be zero after inflation and at the reheating temperature. We prove that by leakage of gravitons into the bulk its density is restored by the freeze-in mechanism, UV dominated and then controlled by the reheating temperature $T_R$ after inflation. Another possibility provided by the same class of models is explored in Section 6. The spectrum of the bulk graviton is a gapped continuum, similar to the case of gapped unparticles. We learned from the case of general unparticle propagators that interaction with ordinary matter can provide self-energies that, when resummed, can give rise to an isolated pole in the second Riemann sheet and with a pole mass near (and below) the mass gap. In the case of the continuum of gravitons, they behave as unparticles with dimension $d_{\mathcal{U}}=3/2$, and couple to the SM fields through their energy-momentum tensors. We have proved that, indeed, an isolated feebly interacting long-lived graviton can appear in the spectrum. We have studied the conditions, from all theoretical and cosmological constraints, under which this isolated state can be cold dark matter. Interestingly enough it is required that a heavy field, beyond the SM, be present with a mass $\sim M_5$. We have proposed such field as the cosmological inflaton localized in the brane. The final result is that the isolated graviton can be DM provided that its mass is below the MeV range. Moreover, as an existence proof, we have presented a simple inflationary model localized in the brane, which satisfies all required properties and with correct predictions of cosmological observables. Interestingly enough, the model evades Lyth’s bound about super-Planckian excursions. A discussion on results and future perspectives is done in Section 7. Finally, for completeness, in the appendices we compare the orbifold and interval pictures, Appendix A, and present results for a general class of $\nu$-models which are asymptotically AdS, in Appendix B.

2. A Class of Dilaton-Gravity Models

We will here consider a slice of 5D space-time between a brane at the value $y_0=0$ in proper coordinates y, the UV boundary brane, and a (possible) admissible singularity [9] placed at $y_s$, a value which has to be determined dynamically. In addition, we will eventually introduce an IR brane, at $y_1<y_s$, responsible for electroweak breaking, where we will assume the SM sector to be localized. The considered interval in the downstairs picture $[0,y_s]$, is completely equivalent to the orbifold $S^1/{\mathbb{Z}}_2$, in the upstairs picture. In the absence of brane terms in the action, even (odd) fields in the orbifold picture have Neumann (Dirichlet) boundary conditions in the interval picture. In this article we will be working in the interval picture. The relation between interval and orbifold pictures being described in Appendix A.

The 5D action $\mathcal{S}$ of the model, including the stabilizing bulk scalar $\varphi$, with mass dimension $3/2$, reads as

$$\begin{array}{ccc}\hfill \mathcal{S}& =& \int d^{4}x{\int }_0^{y_s}dy\sqrt{|det{g}_{MN}|}\left[-{\displaystyle \frac{1}{2{\kappa }^2}}R+{\displaystyle \frac{1}{2}}{g}^{MN}\left({\partial }_M\varphi \right)\left({\partial }_N\varphi \right)-V\left(\varphi \right)\right]\hfill \\ \hfill & -& \sum _a{\int }_{B_a}{d}^{4}x\sqrt{|det{\overline{g}}_{\mu \nu }|}{\lambda }_a\left(\varphi \right)-{\displaystyle \frac{1}{{\kappa }^2}}{\int }_{B_0}{d}^{4}x\sqrt{|det{\overline{g}}_{\mu \nu }|}{K}_0,\hfill \end{array}$$

(1)

where ${\kappa }^2\equiv 1/M_5^3$, $M_5$ being the 5D Planck scale, $V\left(\varphi \right)$ and ${\lambda }_a\left(\varphi \right)$ the bulk and brane potentials of the scalar field $\varphi$, and the index $a=0$, 1 referring to the UV and IR branes, respectively. The IR brane is responsible for the generation of the IR scale ∼ TeV, and contains the brane Higgs potential which spontaneously breaks the electroweak symmetry, thus solving the hierarchy problem between $M_5$ and the TeV scale for the considered value of $A\left(y_1\right)$, as we will see. In Equation (1) the 4D induced metric is ${\overline{g}}_{\mu \nu }=e^{-2A\left(y\right)}{\eta }_{\mu \nu }$, where the Minkowski metric is given by ${\eta }_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$. The last term in Equation (1) is the usual Gibbons-Hawking-York boundary term [10,11], where $K_0$ is the extrinsic UV curvature. In terms of the metric of Equation (2) the extrinsic curvature term reads as [12]$K_0=-4A^{\prime }\left(y_0\right)$. Note that the extrinsic curvature at the singularity is canceled by the action of the determinant.

2.1. No Black Hole background

We will now consider a 5D metric, which generalizes the AdS metric and is given by

$$d{s}^2=g_{MN}d{x}^{M}d{x}^N\equiv e^{-2A\left(y\right)}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }-d{y}^2.$$

(2)

The equations of motion (EoM) read then as

$$\begin{array}{ccc}& & A^{\prime \prime }={\displaystyle \frac{{\kappa }^2}{3}}{\varphi }^{\prime 2}+{\displaystyle \frac{{\kappa }^2}{3}}\sum _a{\lambda }_a\left(\varphi \right)\delta (y-y_a),\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & A^{\prime 2}=-{\displaystyle \frac{{\kappa }^2}{6}}V\left(\varphi \right)+{\displaystyle \frac{{\kappa }^2}{12}}{\varphi }^{\prime 2},\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & {\varphi }^{\prime \prime }=4A^{\prime }{\varphi }^{\prime }+V^{\prime }\left(\varphi \right)+{\displaystyle \sum _{\alpha }}{\lambda }_a^{\prime }\left(\varphi \right)\delta (y-y_a),\hfill \end{array}$$

(5)

where the prime symbol $(\prime )$ will hereafter stand for the derivative of a function with respect to its argument. The EoM in the bulk can also be written in terms of the superpotential $W\left(\varphi \right)$ as [13]

$${\varphi }^{\prime }={\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial W}{\partial \varphi }},A^{\prime }={\displaystyle \frac{{\kappa }^2}{6}}W,$$

(6)

and

$$V\left(\varphi \right)={\displaystyle \frac{1}{8}}{\left({\displaystyle \frac{\partial W}{\partial \varphi }}\right)}^2-{\displaystyle \frac{{\kappa }^2}{6}}{W}^2\left(\varphi \right).$$

(7)

It can be easily checked that if the superpotential W grows faster than ${\varphi }^2$ at large $\varphi$, the profile $\varphi \left(y\right)$ diverges at a finite value $y\equiv y_s$ [14]. Moreover, the Ricci scalar can be written as

$$R={\displaystyle \frac{5}{9}}{\kappa }^4{W}^2\left(\varphi \right)-{\displaystyle \frac{2}{3}}{\kappa }^2{\left({\displaystyle \frac{\partial W}{\partial \varphi }}\right)}^2,$$

(8)

so that the curvature in general diverges at $y=y_s$. The physical interpretation is that spacetime ends at $y_s$.

The brane potential terms in the EoM are responsible for boundary and jumping conditions for the fields in the branes. In particular, by integrating the equations in a neighborhood of each brane, and using Equation (6), we get on the UV boundary

$$W\left(\varphi \left(y_0\right)\right)={\lambda }_0\left(\varphi \left(y_0\right)\right),W^{\prime }\left(\varphi \left(y_0\right)\right)={\lambda }_0^{\prime }\left(\varphi \left(y_0\right)\right),$$

(9)

where the ${\mathbb{Z}}_2$ orbifold conditions have been used. On the other hand, in case there is an IR brane at $y=y_1$ we have to impose continuity conditions for $W\left(\varphi \right)$ and $W^{\prime }\left(\varphi \right)$, i.e.,

$$\Delta W\left(\varphi \left(y_1\right)\right)=0,\Delta W^{\prime }\left(\varphi \left(y_1\right)\right)=0,$$

(10)

where $\Delta X$ is the jump of the function X when crossing the brane. Simple brane potentials satisfying the boundary, Equation (9), and jumping, Equation (10), conditions, and fixing dynamically the values of $\varphi$ at the branes, i.e., $v_a\equiv \varphi \left(y_a\right)$, are given by

$${\lambda }_0\left(\varphi \right)=W\left(\varphi \right)+{\displaystyle \frac{1}{2}}{\gamma }_0{(\varphi -v_0)}^2,{\lambda }_1\left(\varphi \right)={\displaystyle \frac{1}{2}}{\gamma }_1{(\varphi -v_1)}^2.$$

(11)

Integrating the gravitational 5D Lagrangian by parts in the bulk, and after using the background EoM, one can see that there are contributions to the potential localized on the boundaries as $\mp e^{-4A\left(y_a\right)}W$ [15], where the ∓ sign corresponds to the boundary $y_a=(y_0,y_s)$. While this contribution vanishes on the singularity at $y=y_s$, it gives a contribution to the UV boundary such that the effective UV brane potential is

$$U_0\left(\varphi \right)={\lambda }_0\left(\varphi \right)-W\left(\varphi \right)={\displaystyle \frac{1}{2}}{\gamma }_0{(\varphi -v_0)}^2,$$

(12)

which is dynamically minimized for $\varphi =v_0$. Moreover, on the IR brane there is not such boundary contribution and there the effective potential is $U_1\left(\varphi \right)={\lambda }_1\left(\varphi \right)$ which is minimized for the value of $\varphi =v_1$.

For convenience we will define the dimensionless field $\overline{\varphi }\equiv \kappa \varphi /\sqrt{3}$. The properties of the 5D theory depend on the superpotential behavior in the limit $\overline{\varphi }\to \infty$ [14]. In particular, when the asymptotic superpotential behavior is exponential $e^{\nu \overline{\varphi }}$, there are the following possible situations: i) for $\nu <1$ the spectrum is continuous without mass gap, ii) for $\nu >1$ there is a mass gap and a discrete spectrum, and iii) for the critical value $\nu =1$ the spectrum is continuous with a mass gap. These cases can be covered by the superpotential and potential given by

$$W\left(\overline{\varphi }\right)={\displaystyle \frac{6k}{{\kappa }^2}}{e}^{\nu \overline{\varphi }},V\left(\overline{\varphi }\right)=-{\displaystyle \frac{3k^2}{2{\kappa }^2}}(4-{\nu }^2)e^{2\nu \overline{\varphi }}.$$

(13)

We can see that the case $\nu =0$ provides a constant bulk potential and corresponds to the AdS background. The case $\nu =1$ is the so called linear dilaton background, while the limiting case $\nu =2$ and the case $\nu >2$ correspond to a zero and positive bulk potential, respectively. The two cases, $\nu =2$ and $\nu >2$, are excluded as they lead to a bad singularity according to Gubser’s criterium [9].

The solution of the EoM (6) provides the background in proper coordinates, given by

$$\begin{array}{cc}\hfill A\left(y\right)& =-{\displaystyle \frac{1}{{\nu }^2}}log\left(1-{\displaystyle \frac{y}{y_s}}\right),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \overline{\varphi }\left(y\right)& =-{\displaystyle \frac{1}{\nu }}log\left[{\nu }^{2}k(y_s-y)\right],\hfill \end{array}$$

(15)

where we are conventionally fixing the metric at $y=0$ as $A\left(0\right)=0$ 1. The Ricci scalar is

$$R={\displaystyle \frac{4(5-2{\nu }^2)}{{\nu }^4}}{\displaystyle \frac{1}{{(y_s-y)}^2}},$$

(16)

so that the naked singularity at $y=y_s$ becomes manifest.

2.2. Black Hole Background

Let us now consider a generalization of the AdS-Schwarzschild solution, in particular the BH metric of the form 2

$$d{s}_{\mathrm{BH}}^2=g_{MN}d{x}^{M}d{x}^N\equiv e^{-2A\left(y\right)}\left(h\left(y\right)d{t}^2-d{\overrightarrow{x}}^2\right)-{\displaystyle \frac{d{y}^2}{h\left(y\right)}},$$

(17)

where $h\left(y\right)$ is a blackening factor which vanishes at the position of the event horizon, $y=y_h<y_s$. The EoM with this metric read

$$\begin{array}{ccc}& & {\displaystyle \frac{h^{\prime \prime }}{h^{\prime }}}=4A^{\prime },\hfill \end{array}$$

(18)

$$\begin{array}{ccc}& & A^{\prime \prime }={\displaystyle \frac{{\kappa }^2}{3}}{\varphi }^{\prime 2},\hfill \end{array}$$

(19)

$$\begin{array}{ccc}& & A^{\prime 2}={\displaystyle \frac{h^{\prime }}{4h}}{A}^{\prime }-{\displaystyle \frac{{\kappa }^2}{6}}{\displaystyle \frac{V\left(\varphi \right)}{h}}+{\displaystyle \frac{{\kappa }^2}{12}}{\varphi }^{\prime 2},\hfill \end{array}$$

(20)

$$\begin{array}{ccc}& & {\varphi }^{\prime \prime }=\left(4A^{\prime }-{\displaystyle \frac{h^{\prime }}{h}}\right){\varphi }^{\prime }+{\displaystyle \frac{1}{h}}{V}^{\prime }\left(\varphi \right),\hfill \end{array}$$

(21)

where, to simplify the expressions, we have omitted the boundary terms. The solution turns out to be the same as in Equations (14) and (15) with the blackening factor

$$h\left(y\right)=1-{\left({\displaystyle \frac{y_s-y_h}{y_s-y}}\right)}^{{\displaystyle \frac{4-{\nu }^2}{{\nu }^2}}}.$$

(22)

Two of the integration constants have been fixed by the boundary conditions ${lim}_{y\to -\infty }h\left(y\right)=0$ and $h\left(y_h\right)=0$. For $\nu <2$, the condition $0<h\left(y_1\right)<1$ requires $y_1<y_h<y_s$, which then excludes the singularity from the interval, as it is screened by the horizon. The Ricci scalar for the BH metric is

$$R_{\mathrm{BH}}=R+{\displaystyle \frac{3}{{\nu }^2}}{\left({\displaystyle \frac{y_s-y_h}{y_s-y}}\right)}^{{\displaystyle \frac{4-{\nu }^2}{{\nu }^2}}}{\displaystyle \frac{1}{{(y_s-y)}^2}},$$

(23)

where R is the Ricci scalar in the absence of BH, cf. Equation (16). It is clear from the above expressions that the case of no BH is recovered for $y_h\to y_s$.

2.3. Thermodynamic Relations

From Equation (17), the Hawking temperature $T_h$ and the entropy density $s_h$ of the BH can be expressed as

$$\begin{array}{ccc}\hfill T_h& =& {\displaystyle \frac{1}{4\pi }}{e}^{-A\left(y_h\right)}{\left|h^{\prime }\left(y\right)\right|}_{y=y_h}={\displaystyle \frac{1}{4\pi }}\left({\displaystyle \frac{4}{{\nu }^2}}-1\right){\displaystyle \frac{1}{y_s}}{\left(1-{\displaystyle \frac{y_h}{y_s}}\right)}^{{\displaystyle \frac{1-{\nu }^2}{{\nu }^2}}},\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill s_h& =& {\displaystyle \frac{4\pi }{{\kappa }^2}}{e}^{-3A\left(y_h\right)}={\displaystyle \frac{4\pi }{{\kappa }^2}}{\left(1-{\displaystyle \frac{y_h}{y_s}}\right)}^{{\displaystyle \frac{3}{{\nu }^2}}}.\hfill \end{array}$$

(25)

By using these two expressions, we can write the entropy density as a function of $T_h$ as follows

$$s_h\left(T_h\right)={\displaystyle \frac{4\pi }{{\kappa }^2}}{\left[{\displaystyle \frac{4\pi {\nu }^2{y}_s}{4-{\nu }^2}}\right]}^{{\displaystyle \frac{3}{1-{\nu }^2}}}{T}_h^{{\displaystyle \frac{3}{1-{\nu }^2}}}.$$

(26)

The quantity $T_h$ has a key role in the phase transition. To appreciate this, let us consider the case where the SM is either located in the IR brane, or just propagating in the 5D bulk along with bulk gravitons and radions, as a thermal plasma with a temperature T. The temperature T is not necessarily equal to the BH temperature, $T_h$. Then, the model would describe an out-of-equilibrium situation between both subsystems, for which the relevant thermodynamic relations for the internal and free energy densities are [12,17,18]

$$\begin{array}{ccc}\hfill {\rho }_h\left(T_h\right)& =& T_{h}s\left(T_h\right)-{\int }_0^{T_h}s\left({\widehat{T}}_h\right)d{\widehat{T}}_h,\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill f_h(T_h,T)& =& (T_h-T)s\left(T_h\right)-{\int }_0^{T_h}s\left({\widehat{T}}_h\right)d{\widehat{T}}_h.\hfill \end{array}$$

(28)

In fact, by computing $\partial f_h(T_h,T)/\partial T_h$ from Equation (28) it makes manifest that $f_h(T_h,T)$ has a minimum at $T_h=T$ that amounts to

$$f_{\min }=f_h\left(T\right)=-{\int }_0^{T}s\left({\widehat{T}}_h\right)d{\widehat{T}}_h=-{\displaystyle \frac{4\pi }{{\kappa }^2}}{\displaystyle \frac{1-{\nu }^2}{4-{\nu }^2}}{\left[{\displaystyle \frac{4\pi {\nu }^2{y}_s}{4-{\nu }^2}}\right]}^{{\displaystyle \frac{3}{1-{\nu }^2}}}{T}^{{\displaystyle \frac{4-{\nu }^2}{1-{\nu }^2}}},$$

(29)

corresponding to the equilibrium situation. Notice that upon time compactification with periodicity $1/T$ there appears a conical singularity which only cancels at the minimum at $T_h=T$ where the equilibrium is restored [17].

The case $\nu =0$ reproduces the known results from Ref. [17] for which the free energy decreases as $T^4$, as radiation. For the case $0<\nu <1$ it decreases faster than radiation while for $1<\nu <2$ it increases with decreasing temperatures. The case $\nu =1$, corresponding to the linear dilaton background, is ill-defined and will be worked out later on.

3. The Linear Dilaton Background for Particle Physics

We will hereby consider the critical case $\nu =1$ where the superpotential and the bulk potential are

$$W\left(\overline{\varphi }\right)={\displaystyle \frac{6k}{{\kappa }^2}}{e}^{\overline{\varphi }},V\left(\overline{\varphi }\right)=-{\displaystyle \frac{9k^2}{2{\kappa }^2}}{e}^{2\overline{\varphi }},$$

(30)

where $k\lesssim M_5$ is the parameter which determines the 5D curvature. The model defined by the superpotential (30) has a singularity at a finite value of the proper coordinate $y=y_s$, as in soft wall models. It leads to a gapped continuum spectrum, and the hierarchy problem is, more conventionally, solved in the same way as in RS theories, with fundamental scales $M_5$ and k, and a derived TeV scale after warping. The solution for the background in proper coordinates is (cf. Equation (15) 3)

$$\overline{\varphi }\left(y\right)=-log\left[k(y_s-y)\right],A\left(y\right)=-log\left(1-{\displaystyle \frac{y}{y_s}}\right),$$

(31)

and the BH blackening factor

$$h\left(y\right)=1-{\left({\displaystyle \frac{y_s-y_h}{y_s-y}}\right)}^3.$$

(32)

We will also consider in this section two branes, at $y=0$ (the UV boundary) and $y=y_1$ (the IR or Higgs brane) where we assumed the Standard Model, and in particular the Higgs, to be located, such that the values of the IR brane location $y_1$ and the singularity $y_s$ will be dynamically determined by the brane potentials ${\lambda }_a\left(\varphi \right)$, fixing the field $\overline{\varphi }$ at the values ${\overline{v}}_0\equiv \kappa v_0/\sqrt{3}$ and ${\overline{v}}_1\equiv \kappa v_1/\sqrt{3}$ in the UV and IR branes, respectively, such that

$$k{y}_s=e^{-{\overline{v}}_0},k{y}_1=e^{-{\overline{v}}_0}-e^{-{\overline{v}}_1}.$$

(33)

We can then write for the scalar field the background value $\overline{\varphi }\left(y\right)={\overline{v}}_0-log(1-y/y_s)$.

Notice that the first expression in (33) demands that $k{y}_s>0$. Moreover, the solution of the hierarchy problem between $M_5$ and the TeV scale is achieved for a given value of the warp factor at the IR brane $A\left(y_1\right)\equiv A_1$, which imposes the relation

$${\overline{v}}_1-{\overline{v}}_0=A_1.$$

(34)

As we will see the squared mass gap of the continuum graviton and radion spectrum is then given by

$$m_g^2={\displaystyle \frac{9}{4}}{\eta }^2,$$

(35)

with

$$\eta =\pm 1/y_s.$$

(36)

Using that $dz=\pm e^{A\left(y\right)}dy$, where ± corresponds to the sign of $\eta$, the relation between conformally flat and proper coordinates in the model of Equation (30) turns out to be

$$\eta ·(z-z_0)=-log(1-y/y_s),$$

(37)

and the background in these coordinates is given by

$$\overline{\varphi }\left(z\right)=A\left(z\right)+{\overline{v}}_0,A\left(z\right)=\eta ·(z-z_0),$$

(38)

where we fix $z_0\equiv 1/k$. Therefore in conformally flat coordinates the dilaton $\varphi$ is linear and the corresponding model is dubbed as linear dilaton model (LDM).

According with Equation (36) the parameter $\eta$ can have both signs, and accordingly two classes of theories are implemented. The solution with a negative sign of the parameter $\eta$ in (36), i.e., $\eta =-1/y_s$, was considered in Refs. [8,19]. Here we will only consider the solution with $\eta =1/y_s$.

3.1. The $\eta >0$ Case: The Continuum LDM

In this section we will consider the $\eta =1/y_s>0$ case in Equation (36) [20]. The IR brane location in conformally flat coordinates $z_1$ is dynamically determined, as well as the value of $y_1$, by the IR fixing of the dilaton at the value ${\overline{v}}_1$. For simplicity, we will here fix ${\overline{v}}_1=0$ and ${\overline{v}}_0=-A_1<0$. Then, one finds from Equation (33) that $k{y}_s=e^{A_1}$ and $k(y_s-y_1)=1$. The hierarchy problem is then solved by fixing $A_1$ such that $\eta =\mathcal{O}\left(\mathrm{TeV}\right)$, as given by Equation (36), for $k\lesssim M_5$ with

$$\eta =k{e}^{-A_1}.$$

(39)

The value of $M_5$ is determined by the relation of $M_5$ and k with the 4D Planck scale. To evaluate the relationship between $M_5$ and $M_4$ we must consider that the kinetic term of the graviton zero mode has to be canonical. As we will see the zero mode has a constant profile in proper coordinates, i.e., for the theory defined in the interval $[0,y_s]$

$$h_{\mu \nu }^{\left(0\right)}(x,y)=h_0·h_{\mu \nu }\left(x\right)\Rightarrow {\int }_0^{y_s}{e}^{-2A\left(y\right)}{h}_0^{2}dy=1\Rightarrow h_0=\sqrt{3\eta }.$$

(40)

We then obtain 4

$${\kappa }^2{M}_4^2=3{\int }_0^{y_s}{e}^{-2A\left(y\right)}dy\Rightarrow M_5^3=\eta M_4^2,$$

(41)

which yields, for $\eta =\mathcal{O}\left(\mathrm{TeV}\right)$, $M_5\simeq {10}^{13}\mathrm{GeV}\simeq {10}^{-5}{M}_4$, and correspondingly a warp factor $A_1\simeq 23$. In conformal coordinates the location of the branes in units of $\eta$ are at $\eta z_0=e^{-A_1}$ and $\eta z_1=A_1+e^{-A_1}\simeq A_1$, while the singularity is located at infinity, $z_s\to \infty$. The length of the fundamental region is $y_1\simeq y_s\simeq {10}^{-17}$ cm, much larger than the corresponding one in the RS model $\sim {10}^{-31}$ cm, but still much smaller than that in ADD theories [21] $\sim {10}^{-2}$ cm (for the case of two extra dimensions) where only gravity propagates in the bulk.

We can, in principle, consider two kinds of fundamental intervals for our theory:

• When the fundamental interval is $[0,y_1]$, i.e., $[z_0,z_1]$ in conformal coordinates, the theory spectrum is discrete, the first mode mass being $\mathcal{O}\left(\eta \right)$.
• When the fundamental interval is $[0,y_s]$, i.e., $[z_0,\infty )$ in conformal coordinates (as we will consider here), this theory predicts a continuum spectrum with an $\mathcal{O}\left(\eta \right)$ gap: the continuum linear dilaton model (CLDM). This is the interval we will consider in this review where we are focussing on the case $\eta >0$. Notice that this makes the main difference with respect to the $\eta <0$ case considered in Ref. [8] for which gravity decouples ($M_4\to \infty$) when considering the infinite interval, $z_s\to \infty$.

Let us notice that in this theory the warp factor solves the hierarchy problem between the intermediate scale $M_5$ and the TeV scale. This means that the UV completion of this theory should be a string theory with the string mass at the intermediate scale, $M_s\simeq M_5$, thus solving the hierarchy problem between the Planck scale and $M_5$.

3.2. The Gauge Bosons

As we will see in this section, this class of background does not support gauge bosons propagating in the bulk of the extra dimension, unless the gauge bosons KK modes are in the 100 TeV range.

To begin with, let us assume that only the Higgs is localized on the IR brane, while gauge bosons are propagating in the bulk. We will also assume a custodial model based on the bulk gauge group [22,23]

$$SU{\left(3\right)}_c\otimes SU{\left(2\right)}_L\otimes SU{\left(2\right)}_R\otimes U{\left(1\right)}_X,$$

(42)

where $X\equiv B-L$, with 5D gauge bosons $(\mathcal{G},W_L,W_R,X)$, and 5D couplings $(g_c,g_L,g_R,g_X)$ 5, respectively.

The breaking $SU{\left(2\right)}_R\otimes U{\left(1\right)}_X\to U{\left(1\right)}_Y$, where Y is the SM hypercharge with gauge boson B and coupling $g_Y$, is done in the UV brane by boundary conditions. Therefore the gauge fields $(W_L^a,W_R^a,X)$ define $(W_L^a,W_R^{1,2},B,Z_R)$, with (UV, IR) boundary conditions, as

$$\begin{array}{ccc}\hfill & W_L^a(a=1,2,3),\hfill & \hfill (+,+)\end{array}$$

(44)

$$\begin{array}{ccc}\hfill B& ={\displaystyle \frac{g_X{W}_R^3+g_{R}X}{\sqrt{g_R^2+g_X^2}}},\hfill & \hfill (+,+)\end{array}$$

(45)

$$\begin{array}{ccc}\hfill & W_R^{1,2},\hfill & \hfill (-,+)\end{array}$$

(46)

$$\begin{array}{ccc}\hfill Z_R& ={\displaystyle \frac{g_R{W}_R^3-g_{X}X}{\sqrt{g_R^2+g_X^2}}}.\hfill & \hfill (-,+)\end{array}$$

(47)

The $SU{\left(2\right)}_L\otimes SU{\left(2\right)}_R$ symmetry is unbroken in the IR brane, where all composite states are localized, such that the custodial symmetry is exact. After electroweak breaking the different fields are denoted by $(W_L^{\pm },W_R^{\pm },Z_L,Z_R,A_L)$.

The covariant derivative for fermions is

$$\cancel{D}=\cancel{\partial }-i\left[g_L{\displaystyle \sum _{a=1}^3}{\cancel{W}}_L^a{T}_L^a+g_R{\displaystyle \sum _{b=1}^2}{\cancel{W}}_R^b{T}_R^b+g_Y\cancel{B}Y+g_{Z_R}{\cancel{Z}}_R{Q}_{Z_R}\right],$$

(48)

where $g_Y$ and $g_{Z_R}$ are defined in terms of $g_R$ and $g_X$ as

$$g_Y={\displaystyle \frac{g_R{g}_X}{\sqrt{g_R^2+g_X^2}}},g_{Z_R}=\sqrt{g_R^2+g_X^2},$$

(49)

and the hypercharge Y and the charge $Q_{Z_R}$ are defined by

$$Y=T_R^3+Q_X,Q_{Z_R}={\displaystyle \frac{g_R^2{T}_R^3-g_X^2{Q}_X}{g_R^2+g_X^2}},$$

(50)

with $Q_X=(B-L)/2$.

Electroweak symmetry breaking is triggered in the IR brane by the bulk Higgs bi-doublet

$$\mathcal{H}=\left(\begin{array}{cc}{H}_2^0& H_1^{+}\\ H_2^{-}& H_1^0\end{array}\right),Q_X=0,$$

(51)

where the rows transform under $SU{\left(2\right)}_L$ and the columns under $SU{\left(2\right)}_R$. We will denote their VEVs as $\langle H_2^0\rangle \equiv v_2/\sqrt{2}$ and $\langle H_1^0\rangle \equiv v_1/\sqrt{2}$, so that we will introduce the angle $\beta$ as, $cos\beta =v_1/v$ and $sin\beta =v_2/v$, with $v=\sqrt{v_1^2+v_2^2}$.

3.2.1. Massless Gauge Bosons

The Lagrangian is 6

$$\mathcal{L}={\int }_0^{y_s}dy\left[-{\displaystyle \frac{1}{4}}tr{F}_{\mu \nu }{F}^{\mu \nu }-{\displaystyle \frac{1}{2}}{e}^{-2A}tr{A}_{\mu }^{\prime }{A}_{\mu }^{\prime }\right],$$

(52)

where the trace is over gauge indices. After Fourier transforming the coordinates $x^{\mu }$ into momenta $p^{\mu }$ we can make the field decomposition $A_{\mu }(p,y)=f_A\left(y\right)A_{\mu }\left(p\right)/\sqrt{y_s}$, and the EoM of the fluctuations is given by [24]

$$p^2{f}_A+{\displaystyle \frac{d}{dy}}\left(e^{-2A}{f}_A^{\prime }\left(y\right)\right)=0.$$

(53)

In conformal coordinates, and after rescaling the field by $f_A\left(z\right)=e^{A\left(z\right)/2}{\widehat{f}}_A\left(z\right)$, we obtain the Schrödinger like form for the equation of motion

$$-{\ddot{\widehat{f}}}_A\left(z\right)+V_A\left(z\right){\widehat{f}}_A\left(z\right)=p^2{\widehat{f}}_A\left(z\right),$$

(54)

where $\stackrel{.}{X}\equiv dX/dz$ and the effective Schrödinger potential is

$$V_A\left(z\right)={\displaystyle \frac{1}{4}}{\stackrel{.}{A}}^2\left(z\right)-{\displaystyle \frac{1}{2}}\ddot{A}\left(z\right).$$

(55)

Plugging Equation (38) into this equation, we find the following result for the effective potential

$$V_A\left(z\right)={\displaystyle \frac{{\eta }^2}{4}}.$$

(56)

Then, we find the existence of a mass gap of the potential, which translates into a gap followed by a continuum KK spectrum of gauge bosons.

• Green’s functions: We will compute the Green’s functions for gauge bosons propagating in the bulk of the 5D spacetime from y to $y^{\prime }$, where both y and $y^{\prime }$ are considered arbitrary. To compute the Green’s function we have to solve an inhomogeneous version of the equation of motion Equation (53). This is given by

  $$p^2{G}_A(y,y^{\prime };p)+{\displaystyle \frac{d}{dy}}\left(e^{-2A}{G}_A^{\prime }(y,y^{\prime };p)\right)=\delta (y-y^{\prime }),$$

  (57)

  where the prime in $G_A^{\prime }$ denotes derivative with respect to the variable y. One should start from the general solution of this equation, and chose the integration constants such that the corresponding boundary conditions are fulfilled. The Green’s functions are subject to the following boundary and matching conditions

  $$\begin{array}{c}\hfill \Delta G_A\left(y^{\prime }\right)=0,\Delta G_A^{\prime }\left(y^{\prime }\right)=e^{2A\left(y^{\prime }\right)},\Delta G_A\left(y_1\right)=0,\Delta G_A^{\prime }\left(y_1\right)=0,\end{array}$$

  (58)

  in addition to boundary conditions in the UV brane that will be discussed next. In these expressions, only the behavior on the first variable y is shown in the Green’s functions, and we define the jump function as $\Delta f\left(y\right)\equiv {lim}_{ϵ\to 0}\left(f(y+ϵ)-f(y-ϵ)\right)$.
• Gauge bosons with Neumann and Dirichlet boundary conditions: While the SM photon ($A_L$) is subject to Neumann boundary condition in the UV brane, in the considered extension of the SM we will consider Dirichlet boundary conditions on this brane for the extra massless gauge bosons ($W_R^{\pm },Z_R$). Then, the boundary conditions in the UV brane turn out to be

  $$G_A^{(++)\prime }\left(y_0\right)=0,G_A^{(-+)}\left(y_0\right)=0,$$

  (59)

  for the Green’s functions $G_A^{(++)}$ and $G_A^{(-+)}$, respectively. These conditions are supplemented by the other conditions in Equation (58). The results for the IR-brane-to-IR-brane Green’s functions for massless gauge bosons for both $(+,+)$ and $(-,+)$ boundary conditions turn out to be

  $$\begin{array}{ccc}\hfill G_A^{(+,+)}(y_1,y_1;p)& =& \left[{\Delta }_A^{-}{\left({\displaystyle \frac{\eta }{k}}\right)}^{{\delta }_A}-{\Delta }_A^{+}\right]{\displaystyle \frac{{\Delta }_A^{-}}{{\delta }_A}}{\displaystyle \frac{k}{4p^2}},\left(A_L\right),\hfill \end{array}$$

  (60)

  $$\begin{array}{ccc}\hfill G_A^{(-,+)}(y_1,y_1;p)& =& \left[{\left({\displaystyle \frac{\eta }{k}}\right)}^{{\delta }_A}-1\right]{\displaystyle \frac{1}{{\delta }_A}}{\displaystyle \frac{k}{{\eta }^2}},(W_R^{\pm },Z_R),\hfill \end{array}$$

  (61)

  where

  $${\Delta }_A^{\pm }\equiv \pm {\delta }_A-1,{\delta }_A=\sqrt{1-{\displaystyle \frac{4p^2}{{\eta }^2}}}.$$

  (62)
  We have indicated in Equations (60)-(61) the field(s) to which each propagator applies.

3.2.2. Standard Model Massive Gauge Bosons

As we are considering the Higgs sector localized on the IR brane, in the case of the SM massive gauge bosons $A_{\mu }$ (i.e., $W_L$ and $Z_L$) there are extra terms in the 5D Lagrangian, Equation (52), as

$$\Delta {\mathcal{L}}_5=\left(-{\displaystyle \frac{1}{2}}{M}_Z^2{Z}_{\mu }^2-M_W^2{\left|W_{\mu }\right|}^2\right)\delta (y-y_1),M_A^2=y_s{m}_A^2,$$

(63)

(for $A=Z_L,W_L$), which leads to a modification of the equation of motion for the Green’s function, as

$$\left[p^2-y_s{m}_A^2\delta (y-y_1)\right]G_{A,M}(y,y^{\prime };p)+{\displaystyle \frac{d}{dy}}\left(e^{-2A}{G}_{A,M}^{\prime }(y,y^{\prime };p)\right)=\delta (y-y^{\prime }).$$

(64)

Using Equation (64), the derivative of the Green’s functions turns out to be discontinuous at $y=y_1$, with a jump given by 7

$$\Delta G_{A,M}^{\prime }(y_1,y^{\prime })=m_A^2{y}_s{e}^{2A\left(y_1\right)}{G}_{A,M}(y_1,y^{\prime }).$$

(65)

Finally, the results for the IR-brane-to-IR-brane Green’s functions for massive gauge bosons for both $(+,+)$ and $(-,+)$ boundary conditions turn out to be

$$\begin{array}{ccc}\hfill G_A^{(+,+)}(y_1,y_1;m_A;p)& =& {\displaystyle \frac{{\Delta }_A^{-}{\left(\frac{\eta }{k}\right)}^{{\delta }_A}-{\Delta }_A^{+}}{{\delta }_A-k{y}_s{\left(\frac{m_A}{\eta }\right)}^2\left[\frac{{\Delta }_A^{-}}{{\Delta }_A^{+}}{\left(\frac{\eta }{k}\right)}^{{\delta }_A}-1\right]}}{\Delta }_A^{-}{\displaystyle \frac{k}{4p^2}},(W_L^{\pm },Z_L),\hfill \end{array}$$

(66)

$$\begin{array}{ccc}\hfill G_A^{(-,+)}(y_1,y_1;m_A;p)& =& {\displaystyle \frac{{\left(\frac{\eta }{k}\right)}^{{\delta }_A}-1}{{\delta }_A-k{y}_s{\left(\frac{m_A}{\eta }\right)}^2\left[{\left(\frac{\eta }{k}\right)}^{{\delta }_A}-1\right]}}{\displaystyle \frac{k}{{\eta }^2}}.\hfill \end{array}$$

(67)

Notice that these results reproduce in the limit $(m_A\to 0)$ the Green’s functions for massless gauge bosons provided in Section 3.2.1. Notice that the SM massive gauge bosons $(W_L^{\pm },Z_L)$ have $(+,+)$ boundary conditions. For completeness we also enclose the $(-,+)$ boundary conditions for other hypothetical massive gauge fields in the custodial model, although none of these fields are present in our model.

We summarize in Table 1 the boundary conditions of the gauge fields of the custodial model.

3.2.3. Electroweak Precision Observables

When the electroweak symmetry is broken there is a mixing between the SM fields $W_L$ and $Z_L$ and the heavy modes of $W_{L,R}$ and $Z_{L,R}$ induced by the Lagrangian

$$\mathcal{L}=tr|g_L^5{W}_L^a{T}_L^a\mathcal{H}-g_R^5\mathcal{H}{W}_R^a{T}_R^a{|}^2,$$

(68)

where we are indicating with the script $g_5$ the 5D gauge couplings, related to the 4D couplings $g_4$ by $g_5=g_4\sqrt{y_s}$.

After putting the Higgs bidoublet $\mathcal{H}$, which we assume to be localized on the IR brane,

$$\mathcal{H}=\left(\begin{array}{cc}{H}_2^0& H_1^{-}\\ H_2^{-}& H_1^0\end{array}\right)$$

(69)

at its minimum, $\langle H_{1,2}^0\rangle =v_{1,2}$, with $v_1^2+v_2^2=v^2$ and $v=246.22$ GeV, the Lagrangian (68) gives rise to the quadratic terms

$$\begin{array}{cc}\hfill & \mathcal{L}={\displaystyle \frac{v^2}{4}}{y}_s[g_L^2{W}_L(z_1,x)W_L(z_1,x)+g_R^2{W}_R(z_1,x)W_R(z_1,x)-{\displaystyle \frac{2v_1{v}_2}{v^2}}{g}_L{g}_R{W}_L(z_1,x)W_R(z_1,x)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\displaystyle \frac{g_L^2}{c_L^2}}{Z}_L(z_1,x)Z_L(z_1,x)+{\displaystyle \frac{1}{2}}{g}_R^2{c}_R^2{Z}_R(z_1,x)Z_R(z_1,x)-g_L{g}_R{\displaystyle \frac{c_R}{c_L}}{Z}_L(z_1,x)Z_R(z_1,x)],\hfill \end{array}$$

(70)

where $W_X{W}_X\equiv W_X^{-}{W}_X^{+}$ for $X=L,R$, and $W_L{W}_R\equiv W_L^{-}{W}_R^{+}+W_L^{+}{W}_R^{-}$. One has $v_1=v·cos\beta$ and $v_2=v·sin\beta$, and then $2v_1{v}_2/v^2=2t_{\beta }/(1+t_{\beta }^2)$ where we have defined $t_{\beta }\equiv tan\beta$. In the custodial limit $t_{\beta }=1$ and $v_1=v_2=v/\sqrt{2}$.

The fields $W_L$ and $Z_L$ have a zero mode, which is the corresponding SM field, and a gapped continuum of states, while the fields $W_R$ and $Z_R$ do not possess zero mode, and only the continuum of states after the mass gap. For the electroweak observables contributing to the new physics, the oblique T, S and U parameters are defined as [25]

$$\begin{array}{cc}\hfill \alpha T& ={\displaystyle \frac{{\mathsf{\Pi }}_{WW}\left(0\right)}{m_W^2}}-{\displaystyle \frac{{\mathsf{\Pi }}_{ZZ}\left(0\right)}{m_Z^2}},\hfill \\ \hfill \alpha S& =4s_L^2{c}_L^2{\mathsf{\Pi }}_{ZZ}^{\prime }\left(0\right),\hfill \\ \hfill \alpha (S+U)& =4s_L^2{\mathsf{\Pi }}_{WW}^{\prime }\left(0\right).\hfill \end{array}$$

(71)

For the computation of these parameters, we need to select as external fields the zero modes of either $W_L$ and $Z_L$, and only propagate the continuum of states. We will then define the Green’s functions propagating only the continuum of states as

$$\begin{array}{cc}\hfill {\mathcal{G}}_{W_L,Z_L}(p;z_1,z_1)& =G_{W_L,Z_L}(p;z_1,z_1)-G_{W_L,Z_L}^0\left(p\right),\hfill \\ \hfill {\mathcal{G}}_{W_R,Z_R}(p;z_1,z_1)& =G_{W_R,Z_R}(p;z_1,z_1).\hfill \end{array}$$

(72)

By using the notation $\mathcal{G}\left(0\right)\equiv \mathcal{G}(0;z_1,z_1)$, a straightforward calculation yields

$$\alpha T=m_W^2{y}_s\left[{\mathcal{G}}_{W_L}\left(0\right)+{\displaystyle \frac{4t_{\beta }^2}{{(1+t_{\beta }^2)}^2}}{\displaystyle \frac{g_R^2}{g_L^2}}{\mathcal{G}}_{W_R}\left(0\right)\right]-m_Z^2{y}_s\left[{\mathcal{G}}_{Z_L}\left(0\right)+{\displaystyle \frac{g_R^2}{g_L^2}}{c}_L^2{c}_R^2{\mathcal{G}}_{Z_R}\left(0\right)\right],$$

(73)

where $t_{\beta }=v_2/v_1$. Using the results of previous sections we find

$$\begin{array}{cc}\hfill {\mathcal{G}}_{W_L}\left(0\right)& ={\mathcal{G}}_{Z_L}\left(0\right)=-y_s(k{y}_s-2log\left(k{y}_s\right))\equiv {\mathcal{G}}_L\left(0\right),\hfill \\ \hfill {\mathcal{G}}_{W_R}\left(0\right)& ={\mathcal{G}}_{Z_R}\left(0\right)=-y_s(k{y}_s-1)\equiv {\mathcal{G}}_R\left(0\right),\hfill \end{array}$$

(74)

so that

$$\alpha T=m_W^2{y}_s{\displaystyle \frac{s_L^2}{c_L^2}}\left[\left(1-{\displaystyle \frac{1}{s_R^2}}{\displaystyle \frac{{(1-t_{\beta }^2)}^2}{{(1+t_{\beta }^2)}^2}}\right){\mathcal{G}}_R\left(0\right)-{\mathcal{G}}_L\left(0\right)\right].$$

(75)

A similar calculation yields

$$\begin{array}{cc}\hfill \alpha S& =4m_Z^4{y}_s{s}_L^2{c}_L^2\left[{\mathcal{G}}_L^{\prime }\left(0\right)+{\displaystyle \frac{s_L^2{c}_R^2}{s_R^2}}{\mathcal{G}}_R^{\prime }\left(0\right)\right],\hfill \end{array}$$

(76)

$$\begin{array}{cc}\hfill \alpha U& =4m_Z^4{y}_s{s}_L^4{c}_L^2\left[\left(1-{\displaystyle \frac{1}{s_R^2}}{\displaystyle \frac{{(1-t_{\beta }^2)}^2}{{(1+t_{\beta }^2)}^2}}\right){\mathcal{G}}_R^{\prime }\left(0\right)-{\mathcal{G}}_L^{\prime }\left(0\right)\right],\hfill \end{array}$$

(77)

where the prime stands for ${\displaystyle \frac{d}{d{p}^2}}$, and ${\mathcal{G}}_{W_L}^{\prime }\left(0\right)={\mathcal{G}}_{Z_L}^{\prime }\left(0\right)\equiv {\mathcal{G}}_L^{\prime }\left(0\right)$, ${\mathcal{G}}_{W_R}^{\prime }\left(0\right)={\mathcal{G}}_{Z_R}^{\prime }\left(0\right)\equiv {\mathcal{G}}_R^{\prime }\left(0\right)$. Then we find

$$\begin{array}{cc}\hfill {\mathcal{G}}_L^{\prime }\left(0\right)& =-2y_s^3\left(k{y}_s-{log}^2\left(k{y}_s\right)-log\left(k{y}_s\right)-1/2\right),\hfill \\ \hfill {\mathcal{G}}_R^{\prime }\left(0\right)& =-2y_s^3\left(k{y}_s-log\left(k{y}_s\right)-1\right).\hfill \end{array}$$

(78)

We can see that, in the limit of large value of $k{y}_s=e^{A_1}$, ${\mathcal{G}}_{L,R}\left(0\right),{\mathcal{G}}_{L,R}^{\prime }\left(0\right)=\mathcal{O}\left(k{y}_s\right)$, while ${\mathcal{G}}_R\left(0\right)-{\mathcal{G}}_L\left(0\right),{\mathcal{G}}_R^{\prime }\left(0\right)-{\mathcal{G}}_L^{\prime }\left(0\right)=\mathcal{O}{\left(k{y}_s\right)}^0$, which is the cancelation which appears on the observables T and U in the custodial limit $t_{\beta }=1$. In the custodial limit the observables T and U can be written as

$$\begin{array}{cc}\hfill \alpha T& \simeq -2{\left({\displaystyle \frac{m_W}{\eta }}\right)}^2{t}_L^2{A}_1,\hfill \\ \hfill \alpha U& \simeq -8{\left({\displaystyle \frac{m_Z}{\eta }}\right)}^4{s}_L^4{c}_L^2{A}_1^2,\hfill \end{array}$$

(79)

where the custodial cancellation is explicit. However, still the observable S gives a sizable contribution as

$$\alpha S\simeq -8{\left({\displaystyle \frac{m_Z}{\eta }}\right)}^4{s}_L^2{c}_L^2\left(1+{\displaystyle \frac{s_L^2{c}_R^2}{s_R^2}}\right)e^{A_1},$$

(80)

which should be compared with the experimental values given in Ref. [26]

$$\begin{array}{cc}\hfill S_{\exp }& =0.018,{\sigma }_S=0.095,\hfill \\ \hfill T_{\exp }& =0.04,{\sigma }_T=0.12,\hfill \\ \hfill U_{\exp }& =-0.013,{\sigma }_U=0.086,\hfill \end{array}$$

(81)

where we are using the notation $X=X_{\exp }\pm {\sigma }_X$, and correlation coefficients ${\rho }_{ii}=1$, ${\rho }_{ij}={\rho }_{ji}$ with

$${\rho }_{ST}=0.91,{\rho }_{SU}=-0.66,{\rho }_{TU}=-0.88.$$

(82)

Using now the $3\times 3$ covariance C matrix, ${\left(C\right)}_{ij}={\rho }_{ij}{\sigma }_i{\sigma }_j$ one can build the ${\chi }^2$ distribution as

$${\chi }^2=V·C^{-1}·V^T,V=(S_{\exp }-S,T_{\exp }-T,U_{\exp }-U).$$

(83)

For $k\simeq M_5$, we can write the warp factor as $e^{A_1}\simeq {(M_4/\eta )}^{2/3}$. Considering the value of $\alpha \left(\eta \right)$ as given by the one-loop running

$${\displaystyle \frac{1}{\alpha \left(\eta \right)}}={\displaystyle \frac{1}{\alpha \left(m_Z\right)}}-{\displaystyle \frac{16}{3\pi }}log{\displaystyle \frac{\eta }{m_Z}},$$

(84)

we plot in Figure 1 the ${\chi }^2$ distribution in the plane $(\eta ,s_R^2)$. We can see that, independently of the value of the angle $s_R$ the absolute 95% CL lower bound on $\eta$ is ∼100 TeV. This result then excludes, for TeV values of $\eta$, the SM propagating in the bulk, and motivates the localization of the SM on the IR brane. Thus states that propagate in the bulk are the graviton and the radion, apart from possible SM singlets.

3.3. The Graviton Propagator

The graviton is a transverse traceless fluctuation of the metric of the form

$$d{s}^2=e^{-2A\left(y\right)}{g}_{\mu \nu }d{x}^{\mu }d{x}^{\nu }-d{y}^2,g_{\mu \nu }={\eta }_{\mu \nu }+2\kappa h_{\mu \nu }(x,y),$$

(85)

where $h_{\mu }^{\mu }={\partial }_{\mu }{h}^{\mu \nu }=0$. The on-shell quadratic action, in the interval, is given by

$$\mathcal{S}=-{\displaystyle \frac{1}{2}}\int d^{4}x{\int }_0^{y_s}dy{e}^{-2A}\left[{\partial }_{\rho }{h}_{\mu \nu }{\partial }^{\rho }{h}^{\mu \nu }+e^{-2A}{h}_{\mu \nu }^{\prime }{h}^{\prime \mu \nu }\right].$$

(86)

We will use the ansatz $h_{\mu \nu }(x,y)=h\left(y\right)h_{\mu \nu }\left(x\right)$ from where the EoM can be written as

$$e^{2A\left(y\right)}{\left(e^{-4A\left(y\right)}{h}^{\prime }\left(y\right)\right)}^{\prime }+p^{2}h\left(y\right)=0.$$

(87)

In conformal coordinates, cf. Equation (37), and after rescaling the field by $h\left(z\right)=e^{3A\left(z\right)/2}\tilde{h}\left(z\right)$, the EoM for the fluctuation can be written in the Schrödinger like form [14] as

$$-{\tilde{h}}^{\prime \prime }\left(z\right)+V_h\left(z\right)\tilde{h}\left(z\right)=p^2\tilde{h}\left(z\right),$$

(88)

where the potential is given by

$$V_h\left(z\right)={\displaystyle \frac{9}{4}}{A^{\prime }}^2\left(z\right)-{\displaystyle \frac{3}{2}}{A}^{\prime \prime }\left(z\right).$$

(89)

An explicit evaluation of this potential with the expression of $A\left(z\right)$ given by Equation (38) leads to a constant value $V_h\left(z\right)=m_g^2$, where $m_g=3\eta /2$ is the mass gap for the graviton, typical of a continuum of states with a gap.

The interaction with matter localized at the brane $y=y_a$ is found as

$$\delta \mathcal{S}={\displaystyle \frac{1}{2}}\int d^{5}x\sqrt{-g}{T}^{\mu \nu }\delta g_{\mu \nu }\Rightarrow {\mathcal{L}}_{5D}={\displaystyle \frac{1}{M_5^{3/2}}}{T}^{\mu \nu }\left(x\right)h_{\mu \nu }(x,y_a)\delta (y-y_a),$$

(90)

where the energy-momentum tensor is defined as

$$T^{\mu \nu }={\left.{\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\delta \left(\sqrt{-g}{\mathcal{L}}_{5D}\right)}{\delta g_{\mu \nu }}}\right|}_{{\eta }_{\mu \nu }},$$

(91)

and so it is symmetric by definition.

The Green’s function for $h_{\mu \nu }(x,y)$ in the transverse, traceless gauge, is given by [27]

$$D_{\mu \nu ,\rho \sigma }=G_h(y,y^{\prime };p)P_{\mu \nu ,\rho \sigma }\left(p\right),$$

(92)

where

$$P_{\mu \nu ,\rho \sigma }\left(p\right)={\displaystyle \frac{1}{2}}\left({\Theta }_{\mu \rho }{\Theta }_{\nu \sigma }+{\Theta }_{\mu \sigma }{\Theta }_{\nu \rho }\right)-{\displaystyle \frac{1}{3}}{\Theta }_{\mu \nu }{\Theta }_{\rho \sigma },{\Theta }_{\mu \nu }={\eta }_{\mu \nu }-{\displaystyle \frac{p_{\mu }{p}_{\nu }}{p^2}}.$$

(93)

We are using, for the 5D Green’s function $G_h(y,y^{\prime };p)$, a mixed representation where the 4D coordinates $x^{\mu }$ have been Fourier transformed into 4D momenta $p^{\mu }$. After fixing the value of $y^{\prime }$, one can see that the Green’s function obeys the same EoM as the field $h\left(y\right)$, except for an inhomogeneous Dirac delta term, i.e.,

$$p^2{G}_h(y,y^{\prime })+e^{2A\left(y\right)}{\displaystyle \frac{d}{dy}}\left(e^{-4A\left(y\right)}{G}_h^{\prime }(y,y^{\prime })\right)=e^{2A\left(y^{\prime }\right)}\delta (y-y^{\prime }),$$

(94)

where the prime indicates derivative with respect to the variable y and, for simplicity, we are omitting the p dependence from the argument of the Green’s function. After substituting the explicit expression for the background $A\left(y\right)$, we find that the equation writes as

$$G_h^{\prime \prime }(y,y^{\prime })-{\displaystyle \frac{4}{y_s-y}}{G}_h^{\prime }(y,y^{\prime })+{\displaystyle \frac{1}{{(y_s-y)}^2}}{\left({\displaystyle \frac{p}{\eta }}\right)}^2{G}_h(y,y^{\prime })=e^{4A\left(y^{\prime }\right)}\delta (y-y^{\prime }),$$

(95)

whose general solution is given in Ref. [20]. For the particular case where $y\le y^{\prime }$ we find the solution

$$\begin{array}{c}\hfill G_h(y,y^{\prime })={\displaystyle \frac{1}{3\eta }}{\displaystyle \frac{1}{{\delta }_h}}{(1-{\overline{y}}^{\prime })}^{{\displaystyle \frac{3}{2}}{\Delta }_h^{+}}\left(-{(1-\overline{y})}^{{\displaystyle \frac{3}{2}}{\Delta }_h^{-}}+{\displaystyle \frac{{\Delta }_h^{-}}{{\Delta }_h^{+}}}{(1-\overline{y})}^{{\displaystyle \frac{3}{2}}{\Delta }_h^{+}}\right),\end{array}$$

(96)

where we have defined

$${\Delta }_h^{\pm }=\pm {\delta }_h-1,{\delta }_h=\sqrt{1-(4/9)·p^2/{\eta }^2}.$$

(97)

In particular, in the limit $y\to y_0=0$ one finds

$$\begin{array}{ccc}\hfill G_h(y_0,y^{\prime })& =& -{\displaystyle \frac{2}{3}}{\displaystyle \frac{1}{\eta }}{\displaystyle \frac{1}{{\Delta }_h^{+}}}{(1-{\overline{y}}^{\prime })}^{{\displaystyle \frac{3}{2}}{\Delta }_h^{+}}.\hfill \end{array}$$

(98)

We can consider, in particular, the analytical expressions for the IR-to-IR Green’s functions, useful when the SM is localized on the IR brane as in the present model. These are given by

$$\begin{array}{ccc}\hfill G_h(y_1,y_1;p)& =& {\displaystyle \frac{1}{3\eta }}{\displaystyle \frac{e^{3A_1}}{{\delta }_h}}\left(-1+{\displaystyle \frac{{\Delta }_h^{-}}{{\Delta }_h^{+}}}{e}^{-3A_1{\delta }_h}\right).\hfill \end{array}$$

(99)

All Green’s functions include the zero-mode contributions which behave as

$$G_h^0=\underset{p\to 0}{lim}{G}_h(y,y^{\prime };p)={\displaystyle \frac{3\eta }{p^2}},$$

(100)

so that we can define Green’s functions contributed only by the continuum of KK modes, with the zero-mode contribution subtracted out, as

$${\mathcal{G}}_h(y,y^{\prime })\equiv G_h(y,y^{\prime })-G_h^0.$$

(101)

3.4. Coupling of the Graviton with Matter Fields

We are assuming fields located in the brane $y=y_a$. Then, the usual form of the interaction Lagrangian in the 4D effective theory is given by the Lagrangian

$${\mathcal{L}}_{5D}={\displaystyle \frac{1}{M_5^{3/2}}}{T}^{\mu \nu }(x,y)h_{\mu \nu }(x,y)\delta (y-y_a),$$

(102)

where $T_{\mu \nu }(x,y_a)$ is the energy-momentum tensor of the matter fields localized at $y_a$ 8.

In particular, for the SM fields living in the IR brane at $y=y_1$, the energy-momentum tensor is given by

$$T_{\mu \nu }=D_{\mu }{H}^{†}{D}_{v}H+i\overline{\psi }{\gamma }_{\mu }{D}_v\psi -{F_{\mu }}^{\rho }{F}_{\nu \rho }-{\eta }_{\mu \nu }{\mathcal{L}}_{\mathrm{SM}},$$

(103)

where $D_{\mu }$ is the SM covariant derivative, $\psi$ corresponds to all SM left and right-handed fermions and $F_{\mu \nu }$ is the field strength of different gauge fields $F_{\mu \nu }=W_{\mu \nu }^a,B_{\mu \nu }$. The term proportional to ${\eta }_{\mu \nu }$ does not contribute to the different vertices as the tensor $h_{\mu \nu }$ is traceless. Notice that in the broken phase, when $\langle H\rangle =(v/\sqrt{2}){(0,1)}^T$ the massive gauge bosons have contributions to $T_{\mu \nu }$ proportional to $m_V^2{V}_{\mu }{V}_{\nu }$. The graviton zero mode wave function, $h_{\mu \nu }^0(x,y)=h_0\left(y\right)h_{\mu \nu }^0$, where $h_0\left(y\right)=\sqrt{3\eta }$ is canonically normalized as in Equation (40), couples with the energy-momentum tensor at the IR brane as

$${\displaystyle \frac{1}{M_{\mathrm{Pl}}}}{T}^{\mu \nu }(x,y_1)h_{\mu \nu }^0\left(x\right).$$

(104)

We will now consider the coupling with matter of the continuum of KK modes, with Green’s function ${\mathcal{G}}_h(y,y^{\prime })$. The effective field theory (EFT) for matter localized at the brane $y_a$, for momenta $p\ll \eta$, provides the dimension eight operator ${\mathcal{O}}_h(x,y_a)$ with Wilson coefficient $c\left(y_a\right)$ as

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{EFT}}\left(y_a\right)& =c\left(y_a\right){\mathcal{O}}_h(x,y_a),\hfill \\ \hfill {\mathcal{O}}_h(x,y_a)& =T^{\mu \nu }(x,y_a)D_{\mu \nu ,\rho \sigma }{T}^{\rho \sigma }(x,y_a)=T_{\nu }^{\mu }{T}_{\mu }^{\nu }-{\displaystyle \frac{1}{3}}{\left(T_{\mu }^{\mu }\right)}^2,\hfill \end{array}$$

(105)

where the Wilson coefficients here have mass dimension $-4$.

In particular, for the SM which is localized in the IR brane $y=y_1$

$$c\left(y_1\right)=-{\displaystyle \frac{1}{6}}{\displaystyle \frac{1}{{\eta }^4}}.$$

(106)

Thus gravitational interactions are suppressed by the TeV scale $\eta$, reflecting the fact that the continuum of KK modes is localized toward the IR. On the contrary, if there is some extra matter localized in the UV brane, as e.g., SM singlets of a dark sector, the corresponding Wilson coefficient would be

$$c\left(y_0\right)=-{\displaystyle \frac{1}{9}}{\displaystyle \frac{1}{{\eta }^2{M}_4^2}}.$$

(107)

In fact if we define the effective coupling $g_{\mathrm{eff}}\left(y_a\right)$ as

$$|c\left(y_a\right)|\equiv g_{\mathrm{eff}}^2\left(y_a\right){\displaystyle \frac{1}{{\eta }^2}},$$

(108)

we can see that $g_{\mathrm{eff}}\left(y_0\right)\simeq 1/M_4$ while $g_{\mathrm{eff}}\left(y_1\right)\simeq 1/\eta$.

3.5. Low Energy Constraints

The effective Lagrangian in Equation (105) does give rise, in particular, to the dimension eight operators, in the notation of Refs. [28,29],

$${\mathcal{L}}_{\mathrm{EFT}}\supset {\displaystyle \sum _{i=0}^2}{\displaystyle \frac{f_{S_i}}{{\eta }^4}}{\mathcal{O}}_{S_i},$$

(109)

where

$$\begin{array}{cc}\hfill {\mathcal{O}}_{S_0}& =\left(D^{\mu }{H}^{†}{D}^{\nu }H\right)\left(D_{\mu }{H}^{†}{D}_{\nu }H\right),\hfill \\ \hfill {\mathcal{O}}_{S_1}& =\left(D^{\mu }{H}^{†}{D}_{\mu }H\right)\left(D^{\nu }{H}^{†}{D}_{\nu }H\right),\hfill \\ \hfill {\mathcal{O}}_{S_2}& =\left(D^{\mu }{H}^{†}{D}^{\nu }H\right)\left(D_{\nu }{H}^{†}{D}_{\mu }H\right),\hfill \\ \hfill f_{S_0}& =-{\displaystyle \frac{1}{12}},f_{S_1}={\displaystyle \frac{1}{18}},f_{S_2}=-{\displaystyle \frac{1}{12}}.\hfill \end{array}$$

(110)

The contributions of these effective operators to the observables $S,T,U$ have been computed in Ref. [28] as

$$\alpha T=-{\displaystyle \frac{15}{16{\pi }^2}}{(m_W/\eta )}^4\left(f_{S_0}+f_{S_2}+{\displaystyle \frac{2}{5}}{f}_{S_1}\right)(1+c_W^2){\displaystyle \frac{s_W^2}{c_W^2}}log(\eta /m_W),$$

(111)

where $\alpha$ is the fine structure constant, and $s_W\left(c_W\right)$ the sine (cosine) of the electroweak mixing angle ${\theta }_W$, while $S=U=0$. Using now the values in Equation (110) we get

$$\alpha T\simeq {\displaystyle \frac{13}{96{\pi }^2}}{\left(m_W/\eta \right)}^4(1+c_W^2){\displaystyle \frac{s_W^2}{c_W^2}}log(\eta /m_W).$$

(112)

From the experimental bound $T<0.054$, see Ref. [26], one gets the mild bound $\eta \gtrsim 140$ GeV. The small value of the T parameter comes mainly because this effect stems from a dimension eight operator, and is thus suppressed by the fourth power of $1/\eta$.

3.6. High Energy Constraints

The effective Lagrangian in Equation (105) does also give rise to a number of dimension eight operators, which contribute to an anomalous quartic gauge coupling (aQGC) as

$${\mathcal{L}}_{\mathrm{EFT}}\supset \sum _j{\displaystyle \frac{f_{T_j}}{{\eta }^4}}{\mathcal{O}}_{T_j}+\sum _k{\displaystyle \frac{f_{M_k}}{{\eta }^4}}{\mathcal{O}}_{M_k},$$

(113)

where, using the notation of Refs. [28,29], we have

$$\begin{array}{cc}\hfill {\mathcal{O}}_{T_0}& =\left(W^{\mu \nu }{W}_{\mu \nu }\right)\left(W_{\alpha \beta }{W}^{\alpha \beta }\right),{\mathcal{O}}_{T_2}=\left(W^{\mu \alpha }{W}_{\nu \alpha }\right)\left(W_{\mu \beta }{W}^{\nu \beta }\right),\hfill \\ \hfill {\mathcal{O}}_{T_5}& =\left(W^{\mu \nu }{W}_{\mu \nu }\right)\left(B_{\alpha \beta }{B}^{\alpha \beta }\right),{\mathcal{O}}_{T_7}=\left(W^{\mu \alpha }{W}_{\nu \alpha }\right)\left(B_{\mu \beta }{B}^{\nu \beta }\right),\hfill \\ \hfill {\mathcal{O}}_{T_8}& =\left(B^{\mu \nu }{B}_{\mu \nu }\right)\left(B_{\alpha \beta }{B}^{\alpha \beta }\right),{\mathcal{O}}_{T_9}=\left(B^{\mu \alpha }{B}_{\nu \alpha }\right)\left(B_{\mu \beta }{B}^{\nu \beta }\right),\hfill \end{array}$$

(114)

and

$$\begin{array}{cc}\hfill & {\mathcal{O}}_{M_0}=\left(W^{\mu \nu }{W}_{\mu \nu }\right)\left(D^{\alpha }{H}^{†}{D}_{\alpha }H\right),{\mathcal{O}}_{M_1}=\left(W^{\mu \alpha }{W}_{\nu \alpha }\right)\left(D_{\mu }{H}^{†}{D}^{\nu }H\right),\hfill \\ \hfill & {\mathcal{O}}_{M_2}=\left(B^{\mu \nu }{B}_{\mu \nu }\right)\left(D^{\alpha }{H}^{†}{D}_{\alpha }H\right),{\mathcal{O}}_{M_3}=\left(B^{\mu \alpha }{B}_{\nu \alpha }\right)\left(D_{\mu }{H}^{†}{D}^{\nu }H\right),\hfill \end{array}$$

(115)

with Wilson coefficients

$$\begin{array}{cc}\hfill & f_{T_0}={\displaystyle \frac{1}{18}},f_{T_2}=-{\displaystyle \frac{1}{6}},f_{T_5}={\displaystyle \frac{1}{9}},f_{T_7}=-{\displaystyle \frac{1}{3}},f_{T_8}={\displaystyle \frac{1}{18}},f_{T_9}=-{\displaystyle \frac{1}{6}},\hfill \\ \hfill & f_{M_0}=-{\displaystyle \frac{1}{9}},f_{M_1}={\displaystyle \frac{1}{3}},f_{M_2}=-{\displaystyle \frac{1}{9}},f_{M_3}={\displaystyle \frac{1}{3}}.\hfill \end{array}$$

(116)

The LHC constraints on the above operators are obtained from the CMS experiment [30,31,32]. The strongest constraints are over the operators ${\mathcal{O}}_{T_0}$ and ${\mathcal{O}}_{T_2}$ which translate into the 95% CL lower bound $m_g\gtrsim 1.3$ TeV. Projections in FCC-hh, at $\sqrt{s}=100$ TeV and integrated luminosities up to 30 ab⁻¹, have been made on the anomalous $WW\gamma \gamma$ couplings [33] which, for leptonic decay channels of the Ws in the final state, yield future bounds reaching values as $m_g\gtrsim 7$ TeV.

Of course the presence of aQGC induces violation of unitarity, e.g., in longitudinal gauge boson scattering processes involving four vector particles, as the corresponding scattering amplitudes grow with ${\widehat{s}}^2$, where $\sqrt{\widehat{s}}$ is the center-of-mass energy, since the SM cancellation fails. This issue has been generally considered for the operators ${\mathcal{O}}_{S_i}$, ${\mathcal{O}}_{T_j}$ and ${\mathcal{O}}_{M_k}$ in Refs. [29,34]. The unitarity violation indicates a failure of the EFT to describe the corresponding processes at such large values of $\sqrt{\widehat{s}}$. In particular using the general results in Ref. [29] the unitarity constraints imply an upper bound as $\sqrt{\widehat{s}}\lesssim 2m_g$ for the validity of the EFT.

4. Braneworld Cosmology

In this section we will consider a 5D model with just one brane where the SM is located, the UV brane, and the singularity. In this case the graviton interactions with the SM are so feeble that the bulk system (the graviton and radion KK modes) at temperature $T_b$ is never in equilibrium with the SM plasma (which keeps its own temperature $T\ne T_b$). The bulk system will be dubbed as dark sector (DS). After time compactification the bulk system is in equilibrium with the BH, such that $T_h=T_b$, and there is no conical singularity contribution to the free energy from the black hole.

After cosmological inflation by the inflaton field the SM plasma is heated by the inflaton interactions to the reheating temperature $T_R$ while the bulk is not (provided the inflaton only interacts with the SM) so its temperature is much smaller than the SM one $(T_b\ll T_R)$. An energy density for the dark sector can only be provided by freeze-in as we will see.

Here it is convenient to use cosmological coordinates r related to the proper coordinates by

$$1-{\displaystyle \frac{y}{y_s}}={\left({\displaystyle \frac{r}{r_b}}\right)}^{{\nu }^2},\eta ={\displaystyle \frac{1}{{\nu }^2{y}_s}},$$

(117)

where $r_b$ is the location of the brane, which corresponds to the UV brane $y=0$. Then the metric can be written as

$$d{s}^2=g_{MN}d{x}^{M}d{x}^N=e^{-2A\left(r\right)}(h\left(r\right)d{\tau }^2-d{\overrightarrow{x}}^2)-{\displaystyle \frac{e^{-2B\left(r\right)}}{h\left(r\right)}}d{r}^2,$$

(118)

with

$$\begin{array}{cc}\hfill A\left(r\right)& =-log(r/L),\hfill \\ \hfill B\left(r\right)& =-{\nu }^{2}log(r/r_b)+log\left(\eta r\right),\eta =k{e}^{\nu {\overline{v}}_b},\hfill \\ \hfill h\left(r\right)& =1-{\left({\displaystyle \frac{r_h}{r}}\right)}^{4-{\nu }^2},\overline{\varphi }\left(r\right)={\overline{v}}_b-\nu log(r/r_b),\hfill \end{array}$$

(119)

with $v_b$ the value of the field $\varphi$ at the brane as fixed by the brane-localized potential. The $\eta$ parameter is a physical scale, and L is an integration constant with dimension of length that can be fixed to any value. The $h\left(r\right)$ function describes a planar black hole horizon located at $r=r_h\le r_b$ in the bulk. See [35] for a careful treatment of the gauge redundancies and integration constants. Notice that, with respect to the metric (14) we have introduced a non-vanishing constant value at the origin $y=0$: $A\left(r_b\right)$. With this choice $a_b\equiv r_b/L$ will be identified with the scale factor when solving the equations of motion for the Friedmann-Lemaître-Robertson-Walker cosmology in the brane at $r=r_b$.

We can now apply the thermodynamic relations of Section 2.3 in cosmological coordinates, which yield the relations

$$\begin{array}{cc}\hfill T_h& ={\displaystyle \frac{4-{\nu }^2}{4\pi }}\eta {\left({\displaystyle \frac{r_h}{r_b}}\right)}^{1-{\nu }^2},\hfill \end{array}$$

(120)

$$\begin{array}{cc}\hfill s_h& ={\displaystyle \frac{4\pi }{{\kappa }^2}}{\left({\displaystyle \frac{r_h}{r_b}}\right)}^3,\hfill \end{array}$$

(121)

$$\begin{array}{cc}\hfill {\rho }_h& ={\displaystyle \frac{3}{{\kappa }^2}}\eta {\left({\displaystyle \frac{r_h}{r_b}}\right)}^{4-{\nu }^2},\hfill \end{array}$$

(122)

$$\begin{array}{cc}\hfill f_h& =-P_h=-{\displaystyle \frac{1-{\nu }^2}{{\kappa }^2}}\eta {\left({\displaystyle \frac{r_h}{r_b}}\right)}^{4-{\nu }^2}.\hfill \end{array}$$

(123)

Using this result, we can infer important information about the thermodynamical stability of the fluid. We can see that the free energy is negative (positive) for $\nu <1$ ($\nu >1$). This implies that the existence of the fluid, i.e., of the bulk black hole, is thermodynamically favored for $\nu <1$ while it is unfavored for $\nu >1$. Notice however that this conclusion assumes $r_h\ll r_b$. In the $\nu >1$ case, it turns out that the presence of the fluid becomes favored at larger values of $r_h$ [36].

Finally, for $\nu =1$,

$$\begin{array}{cc}\hfill T_h& ={\displaystyle \frac{3}{4\pi }}\eta ,\hfill \end{array}$$

(124)

$$\begin{array}{cc}\hfill s_h& ={\displaystyle \frac{4\pi }{{\kappa }^2}}{\left({\displaystyle \frac{r_h}{r_b}}\right)}^3,\hfill \end{array}$$

(125)

$$\begin{array}{cc}\hfill {\rho }_h& ={\displaystyle \frac{3}{{\kappa }^2}}\eta {\left({\displaystyle \frac{r_h}{r_b}}\right)}^3,\hfill \end{array}$$

(126)

$$\begin{array}{cc}\hfill f_h& =-P_h=0,\hfill \end{array}$$

(127)

so that the free energy vanishes at leading order. Therefore the next-to-leading order term must be determined to assess stability, as done in Ref. [18].

4.1. Braneworld Cosmology and Friedmann Equations

We assume that all the fields of the Standard Model are localized on the brane at $r=r_b$. The brane $T_{\mu \nu }^b$ stress tensor is identified as the SM stress tensor, $T_{\mu \nu }^b\equiv T_{\mu \nu }^{\mathrm{SM}}$. Following Ref. [35], the effective Einstein equation has the form

$$G_{\mu \nu }={\displaystyle \frac{1}{M_4^2}}\left(T_{\mu \nu }^{\mathrm{SM}}+T_{\mu \nu }^h\right)+O\left({\displaystyle \frac{T_{\mathrm{SM}}^2}{M_5^6}}\right),$$

(128)

where the total stress tensor $T_{\mu \nu }^{\mathrm{tot}}=T_{\mu \nu }^{\mathrm{SM}}+T_{\mu \nu }^h$ contains the contribution $T_{\mu \nu }^h$ corresponding to the bulk stress tensor projected on the brane [37,38,39]. The projection procedure leads to the structure of a 4D perfect fluid given by

$$T_{\nu }^{h,\mu }=g^{\mu \lambda }{T}_{\lambda \nu }^h=\mathrm{diag}(-{\rho }_h,P_h,P_h,P_h),$$

(129)

so that we will refer to $T_{\mu \nu }^h$ as the holographic fluid contribution to the stress tensor.

We assume that the brane is moving, so that $r_b$ depends on time. Within the low-energy regime assumed in (128), the brane motion is nonrelativistic and simplifications occur, see [38,39] for details. The effective Einstein equation (128) then produces the effective Friedmann equations

$$3M_4^2{H}^2={\rho }_{\mathrm{tot}},6M_4^2{\displaystyle \frac{{\ddot{r}}_b\left(t\right)}{r_b\left(t\right)}}=-\left({\rho }_{\mathrm{tot}}+3P_{\mathrm{tot}}\right),$$

(130)

where $r_b\left(t\right)/L\equiv a\left(t\right)$ plays the role of the scale factor from the induced metric, with $H={\dot{r}}_b/r_b$ the corresponding Hubble parameter. In these equations, we have ${\rho }_{\mathrm{tot}}\simeq {\rho }_{\mathrm{SM}}+{\rho }_h$ and $P_{\mathrm{tot}}=P_{\mathrm{SM}}+P_h$.

The energy density and pressure for the holographic fluid $({\rho }_h,P_h)$ receive the following contributions

$$\begin{array}{cc}\hfill {\rho }_h^W& =3P_h^W=3\left(1-{\displaystyle \frac{{\nu }^2}{4}}\right){\eta }^2{M}_4^2{\left({\displaystyle \frac{r_h}{r_b}}\right)}^{4-{\nu }^2},\hfill \end{array}$$

(131)

$$\begin{array}{cc}\hfill {\rho }_h^{\varphi }& =-P_h^{\varphi }=-3{\eta }^2{M}_4^2\times \left[1-{\displaystyle \frac{{\nu }^2}{4}}{\left({\displaystyle \frac{r_h}{r_b}}\right)}^{4-{\nu }^2}\right],\hfill \end{array}$$

(132)

$$\begin{array}{cc}\hfill {\rho }_h^{\mathsf{\Lambda }}& =-P_h^{\mathsf{\Lambda }}={\displaystyle \frac{{\mathsf{\Lambda }}_b^2{M}_4^2}{12M_5^6}},\hfill \end{array}$$

(133)

corresponding to projection on the brane of: i) the 5D Weyl tensor; ii) the bulk stress tensor, and; iii) the brane tension, respectively. The summation of these contributions can be split as a vacuum (constant) and a fluid contribution, i.e.,

$${\rho }_h={\rho }_{\mathrm{vacuum}}+{\rho }_{\mathrm{fluid}},P_h=P_{\mathrm{vacuum}}+P_{\mathrm{fluid}}.$$

(134)

The fluid contributions turn out to be

$${\rho }_{\mathrm{fluid}}=3{\eta }^2{M}_4^2{\left({\displaystyle \frac{r_h}{r_b}}\right)}^{4-{\nu }^2},P_{\mathrm{fluid}}=(1-{\nu }^2){\eta }^2{M}_4^2{\left({\displaystyle \frac{r_h}{r_b}}\right)}^{4-{\nu }^2},$$

(135)

while the vacuum contribution is given by

$${\rho }_{\mathrm{vacuum}}=-P_{\mathrm{vacuum}}={\mathsf{\Lambda }}_4{M}_4^2,$$

(136)

where the 4D cosmological constant ${\mathsf{\Lambda }}_4$ and the Planck Mass $M_4$ are related to the parameters of the bulk action as

$${\mathsf{\Lambda }}_4=-3{\eta }^2+{\displaystyle \frac{{\mathsf{\Lambda }}_b^2}{12M_5^6}},M_5^3=\eta M_4^2\sqrt{1+{\displaystyle \frac{{\mathsf{\Lambda }}_4}{3{\eta }^2}}}.$$

(137)

The brane tension can be ultimately tuned to set ${\mathsf{\Lambda }}_4$ to (almost) zero, so that we will consider that the 5D Planck mass turns out to be related to the 4D one by $M_5^3=\eta M_4^2$, a result that is consistent with Equation (42). Notice that these quantities are consistent with those obtained from the BH thermodynamics, Equations (120)-(123).

On the other hand, the SM fields form a thermal bath with temperature T, hence ${\rho }_{\mathrm{SM}}={\rho }_{\mathrm{SM}}\left(T\right)$ and $P_{\mathrm{SM}}=P_{\mathrm{SM}}\left(T\right)$. The brane position today (i.e., at $t=t_0$) is denoted by $r_b\left(t_0\right)=r_{b,0}$. 9 Assuming adiabatic expansion of the universe, we have ${\displaystyle \frac{r_b\left(t\right)}{r_{b,0}}}={\displaystyle \frac{T_0}{T\left(t\right)}}$, where $T_0=0.23$ meV is the temperature of the universe today.

5. Dark Sector as a Holographic Fluid for the LD Background

In the background with $\nu =1$, the pressure of the holographic fluid vanishes up to $\mathcal{O}(r_h^6/r_b^6)$ corrections (see Ref. [18]). In the context of a cosmological braneworld model, we propose the identification [35,38] of the dark sector with a holographic fluid, which can eventually share all properties of dark matter.

The constant temperature of the DS is a hallmark of Hagedorn behavior. We remind that the above thermodynamical properties match those of the thermal state of Little String Theory [6,40]. In LST, the Hagedorn temperature is identified as $T_H={\displaystyle \frac{M_s}{2\pi \sqrt{N}}}$ [35], where $M_s$ is the string scale.

Let us analyze the condition that guarantees that the holographic fluid can account for the dark matter abundance today. Using today’s critical density ${\rho }_{c,0}=3H_0^2{M}_4^2$, the DM abundance is given by the remarkably simple formula

$${\mathsf{\Omega }}_{\mathrm{DM},0}={\left({\displaystyle \frac{\eta }{H_0}}\right)}^2{\left({\displaystyle \frac{r_h}{r_{b,0}}}\right)}^3.$$

(138)

Fixing the dark matter abundance today to its observed value ${\mathsf{\Omega }}_{\mathrm{DM},0}\simeq 0.27$ relates therefore the ratio $r_h/r_{b,0}$ to the $\eta$ parameter. Using today’s value of the Hubble parameter $H_0\simeq 1.44\times {10}^{-42}$ GeV, we obtain a bound on $r_h/r_{b,0}$. The bound can be extended to any time by considering the adiabatic expansion of the universe $(r_b\propto 1/T)$, from which we obtain

$${\displaystyle \frac{r_h}{r_b}}\simeq 3.5\times {10}^{-16}{\displaystyle \frac{T}{\mathrm{GeV}}}{\left({\displaystyle \frac{\mathrm{GeV}}{\eta }}\right)}^{2/3}.$$

(139)

As a result, the horizon is typically very far from the brane, and the $\mathcal{O}(r_h^3/r_b^3)$ corrections relative to the leading orders are tiny, even for the maximum reheating temperature $T_R$ that we will find in Section 5.2.

Having identified the $\nu =1$ holographic fluid as the cosmological dark matter observed in our universe, we should ask about the cosmological history of such a scenario. Then, the following question arises: is there a mechanism capable of producing the holographic fluid with correct abundance in the early universe? In this section we show that a mechanism analogous to freeze-in production [41] naturally occurs in our holographic dark matter framework.

Our main assumption is that the inflaton is brane-localized. As a result, the brane position $r_b$ (i.e., the scale factor) grows exponentially during inflation, such that the bulk can be considered as empty at the end of inflation. It is thus well-motivated to consider the system at the reheating time with no bulk black hole, i.e., $r_h=0$. The corresponding reheating temperature is denoted by $T_R$.

5.1. Heating the Bulk: Graviton Radiation from the Brane

The Standard Model particles on the brane couple to bulk gravitons as

$$\mathcal{L}\supset {\displaystyle \frac{1}{M_5^{3/2}}}{T}_{\mathrm{SM}}^{\mu \nu }\left(x\right)h_{\mu \nu }(z_b,x).$$

(140)

Therefore, processes of the form $\phi +\overline{\phi }\to h_{\mu \nu }$ (where $\phi$ are SM fields) happen in the brane thermal bath. Since the gravitons live in the bulk, this is an emission mechanism through which energy is transferred from the brane to the bulk.

The rate of this energy transfer process, here denoted $\Delta {\dot{\rho }}_{\mathrm{SM}}$, was estimated in [15] and computed in AdS₅ in [42,43,44]. We performed a computation of $\Delta {\dot{\rho }}_{\mathrm{SM}}$ in the LD₅ background using a unitarity cut on the LD propagators computed in [35,45]. The details of the computation are given in Ref. [18]. For center-of-mass energies much higher than the local 5D curvature ${R|}_{\mathrm{brane}}\approx -12{\eta }^2$, the result matches the flat space result (as well as the AdS₅ with negligible curvature, as done in [42,43,44]).

The brane-to-bulk energy transfer rate is

$$\Delta {\dot{\rho }}_{\mathrm{SM}}=-{\displaystyle \frac{2\left({\sum }_{\phi }{g}_{\phi }{\kappa }_{\phi }^2{c}_{\phi }{a}_{\phi }\right)}{5{\pi }^4\eta M_4^2}}\mathsf{\Gamma }\left({\displaystyle \frac{7}{2}}\right)\mathsf{\Gamma }\left({\displaystyle \frac{9}{2}}\right)\zeta \left({\displaystyle \frac{7}{2}}\right)\zeta \left({\displaystyle \frac{9}{2}}\right)T^8,$$

(141)

where $a_s=a_V=1$, $a_f\simeq 0.75$ (s, V and f stand for scalars, vectors and fermions, respectively), and $c_{s,f,V}=(1/12,1/16,1/4)$. Considering the SM degrees of freedom with $g_s=4$, $g_V=12$ and $g_f=45$ we get ${\sum }_{\phi }{g}_{\phi }{\kappa }_{\phi }^2{c}_{\phi }{a}_{\phi }=20.78$, where ${\kappa }_{\phi }$ is the number of spin degrees of freedom of $\phi$ (${\kappa }_s=1$ and ${\kappa }_V={\kappa }_f=2$). Numerically,

$$\Delta {\dot{\rho }}_{\mathrm{SM}}=-C_{\rho }{\displaystyle \frac{T^8}{\eta M_4^2}},C_{\rho }\simeq 3.919.$$

(142)

5.2. Holographic Dark Matter Freeze-In

The 4D effective Einstein equation (128) leads to a 4D effective conservation equation. It can be derived either by contracting with ${\nabla }^{\mu }$ and using the 4D Bianchi equation, or by starting with the 5D conservation equation and projecting it on the brane, see [38,39] for details and checks. The effective conservation equations for the SM and holographic energy densities are

$$\begin{array}{cc}\hfill {\dot{\rho }}_{\mathrm{SM}}& =-4H{\rho }_{\mathrm{SM}}+\Delta {\dot{\rho }}_{\mathrm{SM}},\hfill \end{array}$$

(143)

$$\begin{array}{cc}\hfill {\dot{\rho }}_h& =-3H{\rho }_h+\Delta {\dot{\rho }}_h,\Delta {\dot{\rho }}_{\mathrm{SM}}+\Delta {\dot{\rho }}_h=0.\hfill \end{array}$$

(144)

Since for the holographic fluid ${\rho }_h\propto r_h^3$ (see Equation (126)), the evolution equation (144) can alternatively be thought of as an evolution equation for the $r_h$ parameter. That is, the equation must describe the evolution of the black hole horizon in the bulk, providing a geometric picture of freeze-in. The evolution equation for $r_h$ is found to be

$$9M_4^2{\eta }^2{\displaystyle \frac{{\dot{r}}_h{r}_h^2}{r_b^3}}=\Delta {\dot{\rho }}_h.$$

(145)

In Ref. [18] we showed that this result can be directly derived from the bulk viewpoint.

We assume that our holographic model is valid during inflation. We further assume that the inflaton field is brane-localized. Geometrically, this implies that ${\dot{r}}_b\propto e^{Ht}$, i.e., the brane runs away exponentially towards positive r. As a result of the braneworld inflation, we have $r_h/r_b\sim 0$, i.e., the holographic fluid is inexistent at the end of inflation. In this section we call $r_h\left(T\right)$ the value of the horizon as a function of the temperature, while its constant value at small temperatures (i.e., in standard cosmology) will be denoted as ${\overline{r}}_h$. We do not specify the inflation mechanism, and simply start at the reheating time, with corresponding temperature $T_R$ . 10 The initial condition for the dark matter energy density is

$${\rho }_h\left(T_R\right)=0.$$

(146)

We now will solve the evolution equations (143) and (144). Let us first notice that, for low temperatures, the energy transfer terms $\Delta {\dot{\rho }}_{\mathrm{SM},h}$ are negligible as compared to the other terms in the conservation equations. Because of the initial condition (146), we can also assume that ${\rho }_{\mathrm{SM}}\gg {\rho }_h$ throughout the evolution of the universe, such that the Hubble parameter is controlled by the SM thermal bath,

$$H={\displaystyle \frac{1}{\sqrt{3}{M}_4}}\sqrt{{\rho }_{\mathrm{SM}}+{\rho }_h}\simeq {\displaystyle \frac{1}{\sqrt{3}{M}_4}}\sqrt{{\rho }_{\mathrm{SM}}}.$$

(147)

These points imply that, at low temperatures, the scaling of ${\rho }_{\mathrm{SM}}$ and ${\rho }_h$ from standard cosmology is recovered. We denote the corresponding quantities in standard cosmology with bars,

$${\overline{\rho }}_{\mathrm{SM}}\left(T\right)\propto T^4,{\overline{\rho }}_h\left(T\right)\propto T^3.$$

(148)

Furthermore, we have that $|\Delta {\dot{\rho }}_{\mathrm{SM}}\left(T_R\right)|\ll H{\rho }_{\mathrm{SM}}\left(T_R\right)$ at reheating. Therefore the correction to the SM energy density is small, and we can write

$${\rho }_{\mathrm{SM}}\left(T\right)\approx {\overline{\rho }}_{\mathrm{SM}}\left(T\right)=g_{*}{\displaystyle \frac{{\pi }^2}{30}}{T}^4$$

(149)

to a very good approximation.

Using these simplifications, we now solve the evolution equation (144). The result is

$${\rho }_h\left(T\right)={\overline{\rho }}_h\left(T\right)\times \left(1-{\displaystyle \frac{T^3}{T_R^3}}\right)\mathrm{with}{\overline{\rho }}_h\left(T\right)={\displaystyle \frac{\sqrt{10}{C}_{\rho }}{\pi \eta M_4\sqrt{g_{*}}}}{T}_R^3{T}^3.$$

(150)

This result constitutes the prediction for the dark matter energy density in our holographic model. From (150), we can see that the freeze-in production occurs within roughly one order of magnitude in temperature below $T_R$.

Identifying with (138), the dark matter abundance predicted by freeze-in is found to be

$${\mathsf{\Omega }}_{\mathrm{DM},0}={\displaystyle \frac{\sqrt{10}{C}_{\rho }}{3\pi \sqrt{g_{*}}\eta H_0^2}}{\left({\displaystyle \frac{T_R{T}_0}{M_4}}\right)}^3.$$

(151)

Finally, the energy-to-entropy density ratio is given by

$${\displaystyle \frac{{\rho }_h}{s_{\mathrm{tot}}}}={\displaystyle \frac{45\sqrt{10}{C}_{\rho }}{2{\pi }^3\sqrt{g_{*}}{g}_{*,S}}}{\displaystyle \frac{T_R^3-T^3}{\eta M_4}},$$

(152)

where we used $s_{\mathrm{tot}}\approx s_{\mathrm{SM}}=g_{*,S}{\displaystyle \frac{2{\pi }^2}{45}}{T}^3$ and $g_{*,S}$ is the entropy effective number of degrees of freedom in the SM. This result is shown in the left panel of Figure 2.

The freeze-in production of the dark fluid can equivalently be interpreted as the freeze-in production of the black hole horizon, as given by (145), with solution

$$r_h^3\left(T\right)={\overline{r}}_h^3\times \left(1-{\displaystyle \frac{T^3}{T_R^3}}\right),{\overline{r}}_h^3={\displaystyle \frac{\sqrt{10}{C}_{\rho }}{3\pi \sqrt{g_{*}}}}{\left({\displaystyle \frac{L{T}_0}{\eta M_4}}\right)}^3{T}_R^3,$$

(153)

from where we can see that ${\overline{r}}_h=r_h(T\ll T_R)$. Then (150) can be written as

$${\rho }_h\left(T\right)=3{\eta }^2{M}_4^2{\left({\displaystyle \frac{r_h\left(T\right)}{r_b\left(T\right)}}\right)}^3.$$

(154)

A plot of $r_h/L$ is exhibited in the right panel of Figure 2.

From (151), we relate the reheating temperature to the model parameters. Keeping $M_5$ as the free parameter, we find

$$T_R={\left[{\displaystyle \frac{3\pi {\mathsf{\Omega }}_{\mathrm{DM},0}\sqrt{g_{*}}}{C_{\rho }\sqrt{10}}}\eta H_0^2\right]}^{1/3}{\displaystyle \frac{M_4}{T_0}}\simeq 9.4\times {10}^{-10}{M}_5.$$

(155)

Moreover, considering the bound $M_5\lesssim M_4$ (which corresponds to $\eta \lesssim M_4$) yields for the reheating temperature the global upper bound

$$T_R\lesssim 2.3\times {10}^9\mathrm{GeV}.$$

(156)

This is in good agreement with many realistic inflationary models [47].

Finally, the value of the reheating temperature $T_R$ in (155) needs to satisfy the consistency requirement ${\rho }_{\mathrm{SM}}\left(T_R\right)\lesssim {\eta }^2{M}_4^2$, ensuring the validity of the low-energy approximation used in the effective Einstein equation (128). This translates into the mild (sufficient) condition $M_5\gtrsim 10$ GeV, valid for $T<T_R$.

As expected for freeze-in production, the DM sector is never in thermal equilibrium with the SM as condition (155) guarantees that ${\rho }_h\ll {\rho }_{\mathrm{SM}}$ at temperatures $T\le T_R$ (radiation domination). This also justifies one of the assumptions done in the solving of the evolution equations. The remaining assumption, that $|\Delta {\dot{\rho }}_{\mathrm{SM}}|\ll H{\rho }_{\mathrm{SM}}$ for any $T\le T_R$, is verified explicitly by using the relation (155) in the explicit expression for $\Delta {\dot{\rho }}_h$ given in (142).

6. An Isolated Massive Graviton as Dark Matter

As we have seen, for the LD background, the graviton propagator is a gapped continuum, with a mass gap $m_g=3\eta /2$. It has the same structure as an unparticle propagator [48,49], for which it was proven, after introduction of radiative corrections into the self-energy, to be able to exhibit an isolated pole near (but below) the mass gap [50,51]. The main difference between the graviton and a general unparticle is that for the former the interactions with the SM fields are fixed by the single scale $M_5$ (i.e., $\eta$) and the dimension is also fixed, $d_{\mathcal{U}}=3/2$. In this section we will propose that the isolated massive graviton which appears after the introduction of radiative corrections has the capabilities of being a feebly interacting dark matter candidate [52].

6.1. The Graviton Propagator

We will now study the brane-to-brane graviton propagator in the LD model in the presence of a single brane at $r=r_b$, with the matter located at the brane. This follows similar steps as in Section 3.3, so that we will omit most of the technical details.

The system can also be described in conformal coordinates z, with solution

$$\begin{array}{ccc}\hfill d{s}^2& =& e^{-2A\left(z\right)}\left({\eta }_{\mu \nu }d{\tilde{x}}^{\mu }d{\tilde{x}}^{\nu }-d{z}^2\right),\hfill \end{array}$$

(157)

$$\begin{array}{ccc}\hfill A\left(z\right)& =& \eta (z-z_b),\overline{\varphi }\left(z\right)={\overline{v}}_b+\eta (z-z_b),\eta =k{e}^{{\overline{v}}_b},\hfill \end{array}$$

(158)

where ${\tilde{x}}^{\mu }=a\left(z_b\right)x^{\mu }$ are the 4D physical coordinates, with $a\left(z_b\right)=e^{-\eta z_b}$ the scale factor. The relation with the brane cosmology coordinates is $\eta z=-log(r/L)$. The brane position at $r=r_b$ then corresponds to $z_b=-{\displaystyle \frac{1}{\eta }}log(r_b/L)$, while the singularity at $r_s=0$ corresponds to $z_s=\infty$.

The graviton is a transverse traceless fluctuation of the metric of the form

$$d{s}^2=e^{-2A\left(z\right)}\left[\left({\eta }_{\mu \nu }+2\kappa h_{\mu \nu }(\tilde{x},z)\right)d{\tilde{x}}^{\mu }d{\tilde{x}}^{\nu }-d{z}^2\right],$$

(159)

where we are considering conformal coordinates. If one works with the rescaled field defined by ${\tilde{h}}_{\mu \nu }(\tilde{x},z)=e^{-3A\left(z\right)/2}{h}_{\mu \nu }(\tilde{x},z)$, the corresponding 5D graviton propagator is given by

$$G_{\mu \nu ;\alpha \beta }(z,z^{\prime };p^2)=G_h(z,z^{\prime };p^2)P_{\mu \nu ;\alpha \beta }\left(p^2\right),$$

(160)

where $P_{\mu \nu ;\alpha \beta }\left(p^2\right)$ is given by Equation (93), while the coupling of the graviton $h_{\mu \nu }$ with the SM fields is driven by the conserved energy-momentum tensor $T_{\mathrm{SM}}^{\mu \nu }$, i.e.,

$${\mathcal{L}}_{\mathrm{h}-\mathrm{matter}}={\displaystyle \frac{1}{M_5^{3/2}}}{T}_{\mathrm{SM}}^{\mu \nu }\left(\tilde{x}\right)h_{\mu \nu }(z_b,\tilde{x}).$$

(161)

After solving the equations of motion of the graviton with Neumann boundary condition on the brane ${\partial }_{z}G(z,z^{\prime }){|}_{z=z_b}=0$, and regularity of the solution in the limit $z\to \pm \infty$, one finds that the scalar part of the 5D brane-to-brane graviton propagator is given by [18,20]

$$G_h(z_b,z_b;s)=-{\displaystyle \frac{1}{m_g}}{\displaystyle \frac{1}{\Delta -1}},$$

(162)

where

$$\Delta =\sqrt{1-{\displaystyle \frac{s}{m_g^2}}},\left(s\equiv p^2,m_g={\displaystyle \frac{3}{2}}\eta \right).$$

(163)

There is a continuum of KK graviton modes for $m>m_g$, where $m_g$ is the mass gap. There is also a massless mode corresponding to the physical graviton in 4D. This can be seen by computing the limit of the propagator for $s\to 0$, for which one gets

$$G_h(z_b,z_b;s)\underset{s\to 0}{\simeq }{\displaystyle \frac{2m_g}{s}}.$$

(164)

We will define the graviton propagator with the zero mode subtracted out

$${\mathcal{G}}_h(z_b,z_b;s)\equiv G_h(z_b,z_b;s)-{\displaystyle \frac{2m_g}{s}}=-{\displaystyle \frac{1}{m_g}}{\displaystyle \frac{1}{\Delta +1}}.$$

(164)

In the following we will consider the propagator given by Equation (165) (which does not contain the massless graviton propagator), as the physical graviton mass is protected by diffeomorphism (general coordinate) invariance ($h_{\mu \nu }\to h_{\mu \nu }+{\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$), in much the same way as how gauge invariance protects the photon mass in QED.

6.2. The Isolated Resonance: the Massive Graviton

The brane-to-brane propagator of the massive gravitons was computed in Section 6.1 (see also [18,52]), and is given by

$$G_h\left(s\right)={\displaystyle \frac{-1}{m_g+\sqrt{m_g^2-s}}}.$$

(166)

In this expression for the propagator any graviton zero mode has been subtracted out.

Radiative corrections (see Ref. [52]) generate an isolated pole in the second Riemann sheet of the complex s-plane at $s_p=m_p^2-i{m}_p{\mathsf{\Gamma }}_p$ as the zeros of the equation

$$D_h\left(s_p\right)-{\Sigma }_R\left(s_p\right)-i{\Sigma }_I\left(s_p\right)=0,D_h\left(s\right)=-m_g-\sqrt{m_g^2-s},$$

(167)

with ${\Sigma }_R\equiv \mathrm{Re}\Sigma$ and ${\Sigma }_I\equiv \mathrm{Im}\Sigma$, where $\Sigma$ is the self-energy generated by the states localized in the brane, and $m_p<m_g$.

We will consider the contribution of all states localized on the brane, and simplify the analysis by assuming that the heaviest state contributing to the graviton self-energy has a mass $m\gg m_g$. We will consider such state as a heavy scalar $s=\varphi$, e.g., the inflaton responsible for cosmological inflation (see Ref. [52]). In that case the full value of ${\Sigma }_R$ is dominated by this heaviest state, and in fact ${\Sigma }_R\left(s\right)\simeq {\Sigma }_R^s$ is a constant given by [52]

$${\Sigma }_R^s\simeq -{\displaystyle \frac{1}{32{\pi }^2}}\left(5+4log{\displaystyle \frac{M_5^2}{m^2}}\right){\displaystyle \frac{m^4}{M_5^3}},{\Sigma }_I^s=0,$$

(168)

where m is the inflaton mass, while the lighter SM states coupled to the graviton can contribute to ${\Sigma }_I$.

In Ref. [52] it was found that there is a pole in the second Riemann sheet of the complex s-plane parametrized by $a\in [1,2]$ at

$$m_p=\sqrt{a(2-a)}{m}_g,{\mathsf{\Gamma }}_p=-2\sqrt{{\displaystyle \frac{{(a-1)}^2}{a(2-a)}}}{\Sigma }_I\left(m_p^2\right),$$

(169)

where the case $a=1$ corresponds to $m_p=m_g$, and $a=2$ to $m_p=0$, for a value of the inflaton mass m given by

$${\displaystyle \frac{m\left(a\right)}{M_5}}={\left({\displaystyle \frac{e^{5/2}x}{-\mathcal{W}[-x]}}\right)}^{1/4},\mathrm{with}x={\displaystyle \frac{16{\pi }^{2}a{m}_g}{e^{5/2}{M}_5}}.$$

(170)

$\mathcal{W}$ is the principal branch of the Lambert function, so that we can trade the parameter m by a.

To check the analytical approximation for the solution of the pole equation, we have considered a particular simple model where the real part ${\Sigma }_R$ is led by the heavy scalar $\varphi$ so that we have already incorporated its effect on the propagator

$${\overline{G}}_h\left(s\right)\simeq -{\left(\sqrt{m_g^2-s}-(a-1)m_g+i{\Sigma }_I\right)}^{-1}.$$

(171)

In this expression ${\overline{G}}_h\left(s\right)$ stands for the full graviton propagator including resummed loop contributions.The imaginary part of the self-energy is parametrized as

$${\Sigma }_I\left(s\right)\equiv -\overline{\gamma }{s}^2/m_g^3,$$

(172)

where $\overline{\gamma }$ is an arbitrary dimensionless constant.

The pole mass and width are given by $m_p\left(a\right)=\sqrt{a(2-a)}{m}_g$, so that ${\Sigma }_I\left(m_p^2\right)=-a^2{(2-a)}^2\overline{\gamma }{m}_g$, and ${\mathsf{\Gamma }}_p\left(a\right)=2a^{3/2}{(2-a)}^{3/2}(a-1)\overline{\gamma }{m}_g$. Contour lines of $|{\overline{G}}_h|$, where ${\overline{G}}_h={[G_h^{-1}\left(s\right)-{\Sigma }_R-i{\Sigma }_I\left(s\right)]}^{-1}$, in the plane $s=s_R-i{s}_I$ are shown in Figure 3 for the case $a=1.5$ and $\overline{\gamma }={10}^{-3}$.

The analytical approximation of masses and widths are $m_p^2=0.75m_g^2$ and ${\mathsf{\Gamma }}_p\simeq 6.5\times {10}^{-3}{m}_g$. These values are well reproduced by the pole shown in Figure 3.

The residue of the pole is given by

$$R\left(m_p\right)={\displaystyle \frac{1}{D_h^{\prime }\left(s\right)-{\Sigma }_R^{\prime }{\left(s\right)|}_{m_p^2}}}\simeq 2(a-1)m_g,$$

(173)

and the coupling of the isolated resonance ${\chi }_{\mu \nu }$ to the energy-momentum tensor of matter in the brane is given by

$${\kappa }_{\chi }\equiv {\displaystyle \frac{1}{M_5}}{\left({\displaystyle \frac{R\left(m_p\right)}{M_5}}\right)}^{1/2}\simeq {\displaystyle \frac{\sqrt{3(a-1)}}{M_4}}\equiv {\displaystyle \frac{{\lambda }_{\chi }}{M_4}},$$

(174)

where ${\lambda }_{\chi }$ is the Wilson coefficient and the 4D Planck scale $M_4$ the cutoff, defined in the interval ${\lambda }_{\chi }\in [0,\sqrt{3}]$.

In fact we can use the Wilson coefficient ${\lambda }_{\chi }$ as the free parameter, subject to the constraint $0<{\lambda }_{\chi }<\sqrt{3}$, and the pole mass and width of the isolated resonance are given by

$$\begin{array}{cc}\hfill m_p& =\sqrt{1-{\displaystyle \frac{1}{9}}{\lambda }_{\chi }^4}{m}_g,{\mathsf{\Gamma }}_p=-2{\displaystyle \frac{\frac{{\lambda }_{\chi }^2}{3}}{\sqrt{1-\frac{1}{9}{\lambda }_{\chi }^4}}}{\Sigma }_I\left(m_p^2\right).\hfill \end{array}$$

(175)

In particular for ${\lambda }_{\chi }\ll 1$ we have that $m_p\simeq m_g$ while ${\mathsf{\Gamma }}_p$ is strongly suppressed by $\sim {\lambda }_{\chi }^2$, a property that will be used later on.

6.3. The Isolated Resonance as Dark Matter

Depending on the value of ${\mathsf{\Gamma }}_p$ the resonance can decay on cosmological times, or with a lifetime ${\tau }_p=1/{\mathsf{\Gamma }}_p$ larger than the age of the universe. Only in the latter case the resonance can be a candidate to dark matter. In the former case the resonance will decay and could eventually be detected at future accelerators, or by its indirect effects. Both possibilities depend on the value of ${\mathsf{\Gamma }}_p$ which in turns depends on the value of ${\Sigma }_I\left(m_p\right)$. In our model the width is provided by the imaginary part of $\Sigma$$\left({\Sigma }_I\right)$, which is contributed by the SM fields $\phi$ (see Ref. [52]), such that $m_p^2>4m_{\phi }^2$. 11

There are stringent bounds on decaying DM from indirect production and cosmological observations. In particular DM can be constrained from observations of the galactic and extra-galactic diffuse X-ray and gamma-ray backgrounds yielding upper bounds on ${\tau }_{\chi }$ [53]. Moreover, there are also cosmological constraints on exotic injection of electromagnetic energy and its effect on the CMB power spectra [54,55], and effects of the decay channels $e^{+}{e}^{-}$ and $\gamma \gamma$ in the 21-cm power spectrum [56,57]. All of them fix a global upper limit on the DM lifetime, that we take as ${\tau }_{\chi }\gtrsim {10}^{27}$ s. In Figure 4 we plot contour lines of ${\tau }_{\chi }$ in the plane $({\lambda }_{\chi },m_{\chi })$.

The thick lines correspond to ${\tau }_{\chi }={\tau }_U$, a lifetime equal to the universe lifetime, and ${\tau }_{\chi }={10}^{27}\mathrm{s}$, the lower bound imposed by cosmological observations. Dark matter in the shaded region is then excluded.

The isolated resonance ${\chi }_{\mu \nu }$ then corresponds to a feebly interacting massive particle (FIMP), a massive graviton that in the unshaded region of Figure 4 has a life time much larger than the age of the universe, as ${\tau }_{\chi }>{10}^{27}$ s, and is thus a candidate for the dark matter of the universe. Assuming that the inflaton is mainly coupled to the SM fields (as it is only gravitationally coupled to the massive graviton), after inflation, at the reheating temperature $T_R$, the SM is a plasma in thermal equilibrium, while the FIMP is out of equilibrium with zero density. Its energy density must be produced by the freeze-in mechanism, as it happened with the holographic fluid.

The process $\phi \phi \to \chi$ is too suppressed to yield a sizable energy density given that the inverse process, the decay $\chi \to \phi \phi$, is required to be extremely small for the massive graviton to not decay on cosmological times. The freeze-in production by the 2 to 2 processes ${\phi }_1{\phi }_2\to {\phi }_3\chi$ (where ${\phi }_{1,2,3}$ are SM particles) was studied in great detail in Ref. [58]. It was found that the freeze-in is dominated by the channels involving the QCD coupling, in particular $q\overline{q}\to g\chi$ and $qg\to q\chi$, where g are the gluons, and $q=c,b$, are heavy enough quarks to trigger the effect but light enough to be in thermal equilibrium below the critical temperature $T_c\simeq 160$ GeV of electroweak symmetry breaking [59]. Moreover the resulting energy density is IR dominated (the temperature integral is dominated by temperatures $T\sim T_c$), and thus insensitive to the reheating temperature (but sensitive to the value of the involved quark masses above $T_{\mathrm{QCD}}\simeq 150$ MeV), while the UV contribution is subleading. The final yield gives [52]

$${\mathsf{\Omega }}_{\chi }{h}^2\simeq 5.2\times {10}^{-6}{\lambda }_{\chi }^2{\left({\displaystyle \frac{\mathrm{GeV}}{m_{\chi }}}\right)}^3.$$

(176)

We show in Figure 5 the available region in the plane $({\lambda }_{\chi },m_{\chi })$. The lower shaded region is excluded by overclosure of the universe, as there ${\mathsf{\Omega }}_{\chi }{h}^2>0.12$, while the upper shaded region is excluded by cosmological observations, as there ${\tau }_{\chi }<{10}^{27}$ s.

The range for the isolated resonance, and the inflaton mass m, to be identified as DM is then $m_{\chi }\lesssim 2\mathrm{MeV}\mathrm{and}m\lesssim 4\times {10}^{11}\mathrm{GeV},$ while there is no theoretical lower bound, although Lyman-$\alpha$ forest data [60,61], as well as updated Milky Way satellite counts, and strong gravitational lensing, put experimental lower bounds on warm dark matter (WDM) mass around $m_{\chi }\gtrsim 20$ keV [62], which corresponds to $m\gtrsim 8\times {10}^{10}$ GeV. This yields the allowed bands in the fundamental parameters of our theory that is summarized in Table 2.

6.4. Braneworld Inflation

In the case of braneworld cosmology, where matter fields are confined to the three-brane in the warped 5D space, the Hubble parameter in the brane was computed for the case of the linear dilaton background [18,35,38], as

$$3H^2{M}_4^2={\rho }_b\left(1+{\displaystyle \frac{{\rho }_b}{2\lambda }}\right)+{\rho }_h,\lambda =6M_5^6/M_4^2,{\rho }_h={\displaystyle \frac{\lambda }{2}}{(r_h/r_b)}^3,$$

(177)

where ${\rho }_b$ is the brane energy density, $r_h$ the location of the horizon of the 5D black hole, and ${\rho }_h$, describing a holographic fluid which can play the role of holographic dark matter [18]. As this term scales as matter, its strength will be negligible in the early universe, during inflation, where we expect that ${\rho }_b\gg 2\lambda$, and we can approximate the energy density by the inflaton $\varphi$ potential ${\rho }_b\simeq V\left(\varphi \right)$ so that

$$H^2\simeq {\displaystyle \frac{V}{3M_4^2}}\left(1+{\displaystyle \frac{V}{2\lambda }}\right)\simeq {\displaystyle \frac{V^2}{36M_5^6}}.$$

(178)

The slow-roll parameters for braneworld cosmology have been worked out in Refs. [63,64]. For $V\gg 2\lambda$ they can be written as

$$\begin{array}{cc}\hfill ϵ\left(\varphi \right)& \simeq 12M_5^6{\displaystyle \frac{{\left(V^{\prime }\right)}^2}{V^3}},\hfill \end{array}$$

(179)

$$\begin{array}{cc}\hfill \eta \left(\varphi \right)& \simeq 12M_5^6{\displaystyle \frac{V^{\prime \prime }}{V^2}},\hfill \end{array}$$

(180)

which yield the spectral tilt $n_s=1-6ϵ+2\eta$.

The end of inflation $\varphi ={\varphi }_f$ is fixed by the condition $ϵ\left({\varphi }_f\right)=1$. The beginning of inflation $\varphi ={\varphi }_i$ is related to the number of e-folds before the end of inflation N by

$$N\simeq -{\displaystyle \frac{1}{12M_5^6}}{\int }_{{\varphi }_i}^{{\varphi }_f}{\displaystyle \frac{V^2}{V^{\prime }}}d\varphi .$$

(181)

Finally, the CMB normalization of density perturbations yields

$$A_s^2\simeq {\displaystyle \frac{1}{{\left(360\pi M_5^9\right)}^2}}{\displaystyle \frac{V^6\left({\varphi }_i\right)}{{\left[V^{\prime }\left({\varphi }_i\right)\right]}^2}}\simeq 2.1\times {10}^{-9},$$

(182)

while the ratio of tensor-to-scalar perturbations [64] is given by $r\simeq 24ϵ\left({\varphi }_i\right)$.

In Ref. [52] we studied, as an existence proof, for the case of braneworld (non-standard) cosmology, a simple potential, given by

$$V\left(\varphi \right)=V_0\times \left({\displaystyle \frac{{\left(\frac{\varphi }{\mu }\right)}^2+\alpha }{{\left(\frac{\varphi }{\mu }\right)}^2+1}}-\alpha \right),$$

(183)

where $\alpha <1$ is a real parameter.

In the UV regime, where $\varphi /\mu \gg 1$, we get

$$ϵ\simeq {\displaystyle \frac{96{\mu }^2{M}_5^6}{m^2{\varphi }^6}},\eta \simeq -{\displaystyle \frac{144M_5^6}{m^2{\varphi }^4}},N\simeq {\displaystyle \frac{m^2}{192M_5^6}}\left({\varphi }_i^4-{\varphi }_f^4\right).$$

(184)

In the regime $1\ll {\varphi }_f/\mu \ll {\varphi }_i/\mu$, we get

$${\displaystyle \frac{{\varphi }_f^6}{{\mu }^6}}\simeq {\displaystyle \frac{96M_5^6}{m^2{\mu }^4}},{\displaystyle \frac{{\varphi }_i^4}{{\mu }^4}}\simeq {\displaystyle \frac{192M_5^6}{m^2{\mu }^4}}N,$$

(185)

and the slow-roll parameters at the beginning of inflation ${\varphi }_i$

$$\begin{array}{cc}\hfill {ϵ}_i& \simeq {\displaystyle \frac{1}{16\sqrt{3}}}{\displaystyle \frac{m{\mu }^2}{M_5^3}}{\displaystyle \frac{1}{N^{3/2}}}\Rightarrow r\simeq 24{ϵ}_i\simeq {\displaystyle \frac{\sqrt{3}}{2}}{\displaystyle \frac{m{\mu }^2}{M_5^3}}{\displaystyle \frac{1}{N^{3/2}}},\hfill \\ \hfill {\eta }_i& \simeq -{\displaystyle \frac{3}{4N}}\Rightarrow n_s\simeq 1-{\displaystyle \frac{3}{2N}}.\hfill \end{array}$$

(186)

We have to compare the prediction on the spectral index with the experimental values from Planck [65], $n_s=0.9649\pm 0.0042$ (leading to $N\simeq 43$), from ACT DR6 [66], $n_s=0.974\pm 0.003$ (leading to $N\simeq 58$), and from SPT-3G [67], $n_s=0.9679\pm 0.0033$ (leading to $N\simeq 47$). Moreover, the value of $A_s^2$ is given by

$$A_s^2\simeq {\displaystyle \frac{1}{1800\sqrt{3}{\pi }^2}}{\displaystyle \frac{m^5{\mu }^4}{M_5^9}}{N}^{3/2}\simeq 2.1\times {10}^{-9}.$$

(187)

We can now particularize the previous results to our model, for which the inflaton mass is given in terms of $M_5$ as $m\simeq e^{5/8}{M}_5$, from Equation (170). Then the condition (187) gives

$$\mu \simeq 8.8\times {10}^{-3}{M}_5{\left({\displaystyle \frac{60}{N}}\right)}^{3/8},$$

(188)

and the values of ${\varphi }_{i,f}$, for $N\simeq 60$, are given by

$${\varphi }_f\simeq 0.36M_5,{\varphi }_i\simeq 7.58M_5,$$

(189)

which satisfy the assumed condition $1\ll {\varphi }_f/\mu \ll {\varphi }_i/\mu$, and thus evade the Lyth’s bound [68] on trans-Planckian inflaton excursion. Finally the prediction for the tensor-to-scalar ratio is given by

$$r\simeq 2.7\times {10}^{-7}.$$

(190)

Using now the relation (188) we can easily check that the initial condition $V\left({\varphi }_i\right)\gg 2\lambda$ implies an upper bound on $M_5$ (and so on $m_g$), as

$$M_5\lesssim 3.4\times {10}^{-3}{M}_4\simeq 8.2\times {10}^{15}\mathrm{GeV},$$

(191)

which is widely satisfied by our window in Table 2. Furthermore, the Hubble parameter during inflation is given by

$$H_i=2.3\times {10}^{-5}{M}_5\lesssim 4.5\times {10}^6\mathrm{GeV},$$

(192)

where the last inequality is coming from Table 2. Similarly, the maximum value of the reheat temperature $T_R^{\max }$, assuming good reheating, such that all the inflaton energy $V_i$ is converted into radiation, also gets the upper bound

$$T_R^{\max }\lesssim 0.044M_5\le 8.9\times {10}^9\mathrm{GeV},$$

(193)

which is satisfied by many realistic inflationary models [47].

7. Discussion

In this review we have presented the capabilities of the linear dilaton 5D background from the point of view of particles physics and cosmology.

Application to particle physics consists of models with two branes: the UV and the IR brane, where the Higgs is localized. This is an alternative to theories based on AdS background. In proper coordinates, y, the theory extends from the origin $y=0$ to the singularity at $y_s\equiv 1/\eta$, which thus provides the fundamental scale of the theory $\eta$. For $\eta =\mathcal{O}$ (TeV) the 5D Planck scale is around $M_5\simeq {10}^{-4}{M}_4$ and the warped factor solves the hierarchy between $\eta$ and $M_5$ while the string theory has to provide the solution to the hierarchy between $M_5$ and $M_4$.

The main application of the LD background to cosmology consists in describing unconventional, braneworld cosmology, with a single brane where matter is localized. Under exposure to the Friedmann equations, the location of the brane describes the scale factor of the universe under expansion. Thermodynamical quantities, or equivalently the Gauss-Codazzi equations and Israel junction conditions used to project the energy-momentum tensors on the brane, imply the presence of a pressureless holographic fluid which can play the role of dark matter. It is remarkable the difference with respect to the case of an AdS background where the holographic fluid is dark radiation. Another remarkable property of the LD background is that the graviton propagator is a gapped continuum, unlike the case of the AdS case for which the graviton propagator is an ungapped continuum. Both facts gapped Vs. ungapped continuum and holographic dark matter Vs. holographic dark radiation are connected facts. In both cases the dark energy density is zero after cosmological inflation and is then produced by UV freeze-in.

Moreover for the case of the LD background, the presence of the mass gap in the propagator allows the appearance of an isolated massive graviton, out of the continuum, by resummation of self-energy radiative corrections. Its energy density is also zero after cosmological inflation, as it is feebly coupled to the SM, and generated after the reheating temperature by freeze-in. This provides a new (more conventional) candidate to dark matter that we have proven to satisfy all theoretical, experimental and cosmological constraints in the window [20 keV, 2 MeV]. We have also found a surprising link between the possibility of the massive long-lived graviton as dark matter and the presence of an inflaton localized in the brane with a mass similar to the 5D Planck scale. We have presented, as a proof of existence, a simple model of inflation for the brane cosmology satisfying all cosmological observables, and evading Lyth’s bound.

Future research directions are clearly required to explore the properties and possible experimental detection of the candidates to dark matter we have presented: the holographic fluid and the long-lived isolated massive graviton. As for the experimental prospects, its direct detection seems out of the scope for present facilities, as they are feebly coupled to the SM. More promising are future measurements of deviations of the Newton’s law (fifth force experiments). On the other hand before establishing the holographic fluid as a firm candidate to dark matter the cosmological perturbation theory should be studied in detail. We leave it to future research.