Trans-Planckian quantum corrections and inflationary vacuum fluctuations of non-minimally coupled scalar fields

TL;DR

This paper investigates how trans-Planckian physics influences inflationary vacuum fluctuations and primordial density perturbations, showing that UV effects can significantly modify the inflationary potential under certain conditions.

Contribution

It introduces a method to incorporate trans-Planckian corrections into the inflationary potential using non-minimally coupled scalar fields and adiabatic regularization.

Findings

UV effects are embedded in the effective potential during inflation.

Trans-Planckian corrections can significantly alter the inflationary potential.

Derived a constraint on UV scale: _{UV} bb H/g^{1/2}.

Abstract

In the present paper we discuss how trans-Planckian physics affects inflationary vacuum fluctuations and primordial density perturbations. The trans-Planckian problem during inflation has been widely discussed in literature, but it is still under debate. We reconsider this problem by using the two-point correlation function of the non-minimally coupled scalar fields and constructing the effective potential with the adiabatic (WKB) regularization or approximation. First, we clearly show that the cut-off divergence of the quantum fluctuations does not drastically change during inflation under reasonable assumptions and the corrections can be embedded in standard effective potential. Thus, the UV effects on the primordial density perturbation are well translated into the effective potential. Then, we find out the modified effective potential from the inflationary fluctuations and show how the trans-Planckian or UV corrections change the potential during inflation. We clearly show that the new physics strongly affects the inflation potential during inflation and we obtain a inflationary constraint $\Lambda_{\rm UV} \ll H/g^{1/2}$ where $g$ is the interaction coupling at the UV scale $\Lambda_{\rm UV}$.

Equations

$${\beta }_k=\frac{i{e}^{-2ik{\eta }_0}}{2k{\eta }_0+i}{\alpha }_k,$$

$$\displaystyle\left<{\delta\phi}^{2}\right>\Bigr{|}_{k\approx H}={H}^{2}+{c}_{1}{H}^{2}\left(\frac{{H}^{2}}{\Lambda_{\rm UV}^{2}}\right)+{c}_{2}{H}^{2}\left(\frac{{H}^{2}}{\Lambda_{\rm UV}^{2}}\right)^{2}+{c}_{3}{H}^{2}\left(\frac{{H}^{2}}{\Lambda_{\rm UV}^{2}}\right)^{3}+\cdots$$

$$d{s}^2:=-d{t}^2+a^2\left(t\right){\delta }_{ij}d{x}^{i}d{x}^j,$$

$$\begin{array}{rl}\mathcal{L}\left[\varphi ,S\right]& :=\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi +\frac{1}{2}\left(m_{\varphi }^2+{\xi }_{\varphi }R\right){\varphi }^2+\frac{\lambda }{4}{\varphi }^4\\ & +\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }S{\partial }_{\nu }S+\frac{1}{2}\left(m_S^2+{\xi }_{S}R\right)S^2+\frac{g}{2}{\varphi }^2{S}^2,\end{array}$$

$$\square \varphi +m_{\varphi }^2\varphi +{\xi }_{\varphi }R\varphi +\lambda {\varphi }^3+g{S}^2\varphi =0,$$

$$\square S+m_S^{2}S+{\xi }_{S}RS+g{\varphi }^{2}S=0,$$

$$\square =g^{\mu \nu }{\nabla }_{\mu }{\nabla }_{\nu }=1/\sqrt{-g}{\partial }_{\mu }\left(\sqrt{-g}{\partial }^{\mu }\right)$$

$$\varphi =\varphi \left(\eta ,x\right)+\delta \varphi \left(\eta ,x\right),S=S\left(\eta ,x\right)+\delta S\left(\eta ,x\right)$$

$$m_{\varphi }^2=m_{\varphi }^2\left(\mu \right)+\delta m_{\varphi }^2,{\xi }_{\varphi }={\xi }_{\varphi }\left(\mu \right)+\delta {\xi }_{\varphi },\lambda =\lambda \left(\mu \right)+\delta \lambda ,g=g\left(\mu \right)+\delta g,$$

$$\square \varphi +\left(m_{\varphi }^2\left(\mu \right)+\delta m_{\varphi }^2\right)\varphi +\left({\xi }_{\varphi }\left(\mu \right)+\delta {\xi }_{\varphi }\right)R\varphi$$

$$+3\left(\lambda \left(\mu \right)+\delta \lambda \right)⟨\delta {\varphi }^2⟩\varphi +\left(\lambda \left(\mu \right)+\delta \lambda \right){\varphi }^3+\left(g\left(\mu \right)+\delta g\right)⟨\delta S^2⟩\varphi =0,$$

$$\left(\square +m_{\varphi }^2\left(\mu \right)+{\xi }_{\varphi }\left(\mu \right)R+3\lambda \left(\mu \right){\varphi }^2+g\left(\mu \right)S^2\right)\delta \varphi =0,$$

$$\delta \varphi \left(\eta ,x\right)=\int d^{3}k\left(a_k\delta {\varphi }_k\left(\eta ,x\right)+a_k^{†}\delta {\varphi }_k^{\ast }\left(\eta ,x\right)\right),$$

$$⟨0|\delta {\varphi }^2|0⟩={\int }_0^{{\Lambda }_{\text{UV}}}{d}^{3}k{\left|\delta {\varphi }_k\left(\eta ,x\right)\right|}^2=\frac{1}{2{\pi }^2{a}^2\left(\eta \right)}{\int }_0^{{\Lambda }_{\text{cut-off}}}dk{k}^2{\left|\delta {\chi }_k\left(\eta \right)\right|}^2,$$

$${\delta \chi }_k^{\prime \prime }\left(\eta \right)+{\Omega }_k^2\left(\eta \right){\delta \chi }_k\left(\eta \right)=0,$$

$${\Omega }_k^2\left(\eta \right)=k^2+a^2\left(\eta \right)\left(m_{\varphi }^2+g{S}^2+3\lambda {\varphi }^2+\left({\xi }_{\varphi }-1/6\right)R\right)=k^2+a^2\left(\eta \right)M^2\left(\varphi \right).$$

$$\delta {\chi }_k\left(\eta \right)=\frac{1}{\sqrt{2{\Omega }_k\left(\eta \right)}}\left\{{\alpha }_k\left(\eta \right){\delta \phi }_k\left(\eta \right)+{\beta }_k\left(\eta \right){\delta \phi }_k^{\ast }\left(\eta \right)\right\},$$

$$\displaystyle\left<{\delta\phi}^{2}\right>=\frac{1}{4{\pi}^{2}a^{2}\left(\eta\right)}\int_{0}^{\Lambda_{\textrm{cut-off}}}dk{k}^{2}{\Omega}_{k}^{-1}\biggl{\{}1+2{\left|{\beta}_{k}\right|}^{2}+{\alpha}_{k}{\beta}_{k}^{*}{\delta\varphi}_{k}^{2}+{\alpha}_{k}^{*}{\beta}_{k}{\delta\varphi^{*}_{k}}^{2}\biggr{\}}.$$

$$⟨{\delta \varphi }^2⟩={⟨{\delta \varphi }^2⟩}^{(\mathrm{q})}+{⟨{\delta \varphi }^2⟩}^{(\mathrm{c})},$$

$${⟨{\delta \varphi }^2⟩}^{(\mathrm{q})}$$

$${⟨{\delta \varphi }^2⟩}^{(\mathrm{c})}$$

$$\begin{array}{rl}{⟨{\delta \varphi }^2⟩}^{(\mathrm{c})}& =\int \frac{dk}{k}{P}_{\delta \varphi }\left(\eta ,k\right)\\ & =⟨{\delta \varphi }^2⟩-{⟨{\delta \varphi }^2⟩}^{(\mathrm{q})}=\frac{1}{4{\pi }^2{a}^2\left(\eta \right)}{\int }_0^{{\Lambda }_{\text{cut-off}}}dk{k}^2{\Omega }_k^{-1}\left\{2n_k+2\mathrm{Re}{z}_k\right\}\\ & =\frac{1}{4{\pi }^2{a}^2\left(\eta \right)}\left[{\int }_0^{{\Lambda }_{\text{cut-off}}}2dk{k}^2{\left|\delta {\chi }_k\right|}^2-{\int }_0^{{\Lambda }_{\text{cut-off}}}dk{k}^2{\Omega }_k^{-1}\right],\end{array}$$

$$\delta {\chi }_k\left(\eta \right)=\sqrt{\frac{\pi }{4}}{\eta }^{1/2}\left\{{\alpha }_k{H}_{\nu }^{(2)}\left(k\eta \right)+{\beta }_k{H}_{\nu }^{(1)}\left(k\eta \right)\right\},$$

$$\nu \equiv \sqrt{\frac{9}{4}-\frac{M^2\left(\varphi \right)}{H^2}}\simeq \frac{3}{2}-\frac{M^2\left(\varphi \right)}{3H^2},$$

$${\alpha }_k=$$

$${\beta }_k=$$

$${⟨{\delta \varphi }^2⟩}^{(\mathrm{c})}$$

$$=\frac{{\eta }^2{H}^2}{2{\pi }^2}{\int }_0^H{k}^2{\left|\delta {\chi }_k\right|}^{2}dk+\frac{{\eta }^2{H}^2}{2{\pi }^2}{\int }_H^{\sqrt{2}/|\eta |}{k}^2{\left|\delta {\chi }_k\right|}^{2}dk.$$

$$\displaystyle\lim_{\Lambda_{\textrm{cut-off}}\rightarrow\infty}\frac{1}{4{\pi}^{2}a^{2}\left(\eta\right)}\Biggl{[}\int_{\sqrt{2}/\left|\eta\right|}^{\Lambda_{\textrm{cut-off}}}{2k^{2}{\left|\delta{\chi}_{k}\right|}^{2}dk}-\int_{\sqrt{2}/\left|\eta\right|}^{\Lambda_{\textrm{cut-off}}}{dk{k}^{2}{\Omega}_{k}^{-1}}\Biggr{]},$$

$$H_{\nu }^{(1,2{)}^{\prime }}\left(k{\eta }_0\right)=H_{\nu -1}^{(1,2)}\left(k{\eta }_0\right)-\frac{\nu }{k{\eta }_0}{H}_{\nu }^{(1,2)}\left(k{\eta }_0\right),$$

$$\left|{\alpha }_k-{\beta }_k\right|=\sqrt{\frac{\pi k}{2H}}\left|J_{\nu -1}\left(k{\eta }_0\right)+\left(i-\frac{H}{2k}+\frac{\nu H}{k}\right)J_{\nu }\left(k{\eta }_0\right)\right|.$$

$$J_{\nu }\left(k{\eta }_0\right)$$

$$H_{\nu }^{(2)}\left(k{\eta }_0\right)$$

$${\left|\delta {\chi }_k\right|}^2$$

Full text

I Introduction

Inflation STAROBINSKY198099 ; Guth:1980zm ; Sato:1980yn ; Linde:1981mu ; Albrecht:1982wi is the most standard cosmological paradigm describing the early Universe and there are accumulating observational evidences which support its existence (see e.g. Ref. Akrami:2018odb ). Famously, the inflation assumes a quasi de Sitter expansion and explains for various initial condition problems of the standard big bang cosmology elegantly. However, one of the most attractive aspects of inflation is that the quantum fluctuations of the scalar fields during inflation lead to the primordial density perturbations Mukhanov:1981xt ; Hawking:1982cz ; Guth:1982ec ; Starobinsky:1982ee ; Mukhanov:1990me . The quantum fluctuations are frozen after the wavelength becomes larger than the Hubble radius, and finally become the classical density perturbations. The primordial inhomogeneities generate large-scale structure of the Universe and provide the highly precise test of the inflation with the cosmic microwave background (CMB) anisotropies.

The inflationary trans-Planckian problem Niemeyer:2000eh ; Brandenberger:2000wr ; Martin:2000xs ; Tanaka:2000jw ; Starobinsky:2001kn ; Starobinsky:2002rp ; Hui:2001ce ; Niemeyer:2001qe ; Kaloper:2002uj ; Kaloper:2002cs ; Burgess:2002ub ; Brandenberger:2002hs ; Elgaroy:2003gq ; Greene:2004np ; Greene:2005wk ; Danielsson:2002kx ; Danielsson:2002mb ; Danielsson:2002qh ; Danielsson:2005cc ; Danielsson:2006gg ; Goldstein:2002fc ; Easther:2002xe ; Chung:2003wn ; Kaloper:2003nv ; Burgess:2003hw ; Alberghi:2003am ; Martin:2003kp ; Meerburg:2010rp ; Kundu:2011sg ; Groeneboom:2007rf ; Ashoorioon:2013eia ; Ashoorioon:2014nta ; Ashoorioon:2017toq ; Broy:2016zik suggests that inflationary perturbations provide important clues about trans-Planckian or ultraviolet (UV) physics, which has been widely discussed and still under debate. In standard paradigm of the inflation we usually assume that the quantum fluctuations are valid up to an infinite short length. But, it is not realistic since new physics is expected to be below the Planck scale and this simple assumption would be incorrect. There has been many discussions about how trans-Planckian or UV physics affect inflationary perturbations and leave a specific imprint on the CMB. Naively, the inflationary perturbations are not drastically affected by the trans-Planckian physics as far as $H\approx\Lambda_{\rm UV}\approx M_{\rm P}=\sqrt{G}$ where $H$ is the Hubble constant parameter or unless the Lorentz invariance is broken (see e.g. Ref. Starobinsky:2001kn ; Starobinsky:2002rp ). However, even if these effects are sufficiently small, it could provide important clues about the high-energy scale physics. For instance, taking the initial vacuum as $\alpha$-vacua defined at the finite time Allen:1985ux ; Mottola:1984ar , the Bogoliubov coefficients ${\alpha}_{k}$ and ${\beta}_{k}$ are constrained by the following relation,

[TABLE]

where $\eta_{0}$ is the conformal initial time and $k$ is the wave mode. Note that the Bunch-Davies vacuum is restored in the infinite past (${\eta}_{0}\rightarrow-\infty$, ${\alpha}_{k}=1$ and ${\beta}_{k}=0$). Initial condition should be imposed when the wavelength crosses to some fundamental length scale. Thus, the initial condition could be imposed at the $k$-dependent initial time $\eta_{0}=-\Lambda_{\rm UV}/Hk$ where $\Lambda_{\rm UV}$ is the cut-off scale. On the other hand, based on the effective field theory approach, the inflationary fluctuation can be written as follows Kaloper:2002uj ; Kaloper:2002cs :

[TABLE]

where the coefficients ${c}_{i}$ are determined by the UV physics and $\mathcal{O}\left(H^{2}/\Lambda_{\rm UV}^{2}\right)$ matches the standard effective field theory approach which introduces the higher-dimensional operators and capture the influence of the UV physics. There is various estimation of the UV corrections of the inflationary fluctuation on the CMB power spectrum, and various discussion to explore physics at very short distance scales from the CMB measurements.

In this paper we discuss how the UV or trans-Planckian physics affects inflationary vacuum fluctuations and primordial density perturbations. 111 The trans-Planckian problems should be discussed in the framework of quantum gravity (QG). However, there is no consistent theory DeWitt:2007mi ; Smolin:2003rk ; Giddings:2011dr due to the non-renormalizable or non-unitary properties. Thus, it is natural to adopt the semiclassical approach to the gravity birrell1984quantum ; fulling1989aspects ; parker2009quantum ; DeWitt:1975ys ; Bunch:1979uk ; buchbinder1980effective ; Shapiro:2008sf . First, we consider the two-point correlation function $\left<{\delta\phi}^{2}\right>$ of the non-minimally coupled scalar fields using the adiabatic (WKB) regularization or approximation and clearly show that the cut-off divergence of the quantum fluctuations does not drastically change during inflation under. Then, the quantum corrections can be embedded in the effective potential and therefore, the UV effects on the primordial density perturbation are well translated into the effective potential. Then, we construct modified effective potential from the inflationary fluctuations and show how the trans-Planckian or UV corrections change the potential during inflation. We demonstrate that the new physics drastically changes the inflationary perturbations.

II The UV divergence of quantum fluctuations

The quantum fluctuation necessarily causes a problem about renormalization. In quantum field theory (QFT), the two-point correlation function $\left<{\delta\phi}^{2}\right>$ which express the quantum fluctuation have the UV (quadratic and logarithmic) divergences and therefore some regularization or renormalization methods are required. In flat spacetime the divergences of the quantum fluctuation can be eliminated by the bare parameters of the Lagrangian thorough the standard renormalization technique. However, in curved spacetime birrell1984quantum due to the quantum particle creations the representation of the quantum fluctuation and the renormalization have some ambiguity. In this section, let us revisit the renormalization of the quantum fluctuation in de Sitter spacetime using the adiabatic (WKB) regularization or approximation Zeldovich:1971mw ; bunch1980adiabatic ; Parker:1974qw ; Fulling:1974pu ; Fulling:1974zr ; birrell1978application ; Anderson:1987yt ; Haro:2010zz ; Haro:2010mx ; Kohri:2017iyl which is a powerful method to remove the divergences. Consequently, we found out that the inflationary fluctuation or particle creation during inflation are sequestered from the UV or trans-Planckian physics.

Through this paper, we consider a spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime,

[TABLE]

where $a\left(t\right)$ is the scale factor and the scalar curvature is give as $R=6{\left({\dot{a}}/{a}\right)}^{2}+6\left({\ddot{a}}/{a}\right)=6\left({a^{\prime\prime}}/{a^{3}}\right)$ where $\eta$ is the conformal time and defined by $d\eta=dt/a$. The scale factor becomes $a\left(t\right)=e^{Ht}$ and the scalar curvature is expressed as $R=12H^{2}$ in de Sitter spacetime.

We assume two non-minimal coupled scalar fields with the interaction coupling $g$ where inflaton field is $\phi$, massive scalar field is $S$ with $m_{S}\sim\Lambda_{\rm UV}\gg m_{\phi}$. The bare Lagrangian is defined by,

[TABLE]

where $\xi_{\phi},\xi_{S}$ are the non-minimal curvature couplings. The Klein-Gordon equations for these two scalar field are given as follows:

[TABLE]

where

[TABLE]

which express generally covariant d’Alembertian operator. Next, we treat scalar fields $\phi$, $S$ as the field operators acting on the ground states and these scalar fields $\phi$, $S$ are decomposed into the classic parts and the quantum parts;

[TABLE]

and we assume $\left<0\right|{\delta\phi\left(\eta,x\right)}\left|0\right>=\left<0\right|{\delta S\left(\eta,x\right)}\left|0\right>=0$. Introducing the renormalized parameters and counter-terms to be

[TABLE]

the one-loop Klein-Gordon equations of the inflaton field are written as

[TABLE]

where the first equation shows the dynamics of the average inflaton field, whereas the second equation shows the quantum fluctuation of the inflaton field.

The quantum field $\delta\phi$ can be decomposed into each $k$ modes by

[TABLE]

where the creation and annihilation operators of $\delta{\phi}_{k}$ are required to satisfy the standard commutation relations \bigl{[}{a}_{k},{a}_{k^{\prime}}\bigr{]}=\bigl{[}{a}_{k}^{\dagger},{a}_{k^{\prime}}^{\dagger}\bigr{]}=0 and \bigl{[}{a}_{k},{a}_{k^{\prime}}^{\dagger}\bigr{]}={\delta}\left(k-k^{\prime}\right). The in-vacuum state $\left|0\right>$ is defined by $a_{k}\left|0\right>=0$ and corresponds to the initial conditions of the mode functions of $\delta{\phi}_{k}$. The quantum fluctuation $\left<{\delta\phi}^{2}\right>$ of the inflaton field can be written by 222 Note that $\Lambda_{\textrm{cut-off}}$ with the rescaled mode functions $\delta{\chi}_{k}\left(\eta\right)$ corresponds to $a\left(\eta\right)\Lambda_{\rm UV}$.,

[TABLE]

where we introduce the rescaled mode functions $\delta{\chi}_{k}\left(\eta\right)$ as ${\delta\phi}_{k}\left(\eta,x\right)=e^{ik\cdot x}\delta{\chi}_{k}\left(\eta\right)/{\left(2\pi\right)}^{3/2}a\left(\eta\right)$ From Eq. (10), the Klein-Gordon equation for the quantum rescaled field $\delta\chi$ is given by

[TABLE]

where

[TABLE]

Now, we rewrite the rescaled mode function $\delta\chi\left(\eta\right)$ by the Bogoliubov coefficients ${\alpha}_{k}\left(\eta\right)$, ${\beta}_{k}\left(\eta\right)$ as

[TABLE]

where ${\alpha}_{k}\left(\eta\right)$, ${\beta}_{k}\left(\eta\right)$ satisfy the Wronskian condition: ${\left|{\alpha}_{k}\left(\eta\right)\right|}^{2}-{\left|{\beta}_{k}\left(\eta\right)\right|}^{2}=1$. The initial conditions for ${\alpha}_{k}\left(\eta_{0}\right)$, ${\beta}_{k}\left(\eta_{0}\right)$ are equivalent to the choice of the in-vacuum state. From Eq. (14) the quantum fluctuation $\left<{\delta\phi}^{2}\right>$ can be given by Ringwald:1987ui

[TABLE]

For convenience, we introduce the following quantities $n_{k}={\left|{\beta}_{k}\right|}^{2}$ and $z_{k}={\alpha}_{k}{\beta}_{k}^{*}{\delta\varphi}_{k}^{2}$ where $n_{k}={\left|{\beta}_{k}\left(\eta\right)\right|}^{2}$ can be interpreted as the particle number density created in curved spacetime. Using $n_{k}$ and $z_{k}$, we obtain the following expression of the quantum fluctuation of the inflaton field as

[TABLE]

where:

[TABLE]

where the first expression can be regarded as the classic fluctuations and expresses particle creations in curved spacetime birrell1984quantum whereas the second expression obviously diverges and is consistent with the flat spacetime. Therefore, we expect that the first expression is finite and insensitivity to the cut-off parameter $\Lambda_{\textrm{cut-off}}$.

Let us discuss the issues using the adiabatic (WKB) regularization method Zeldovich:1971mw ; bunch1980adiabatic ; Parker:1974qw ; Fulling:1974pu ; Fulling:1974zr ; birrell1978application ; Anderson:1987yt ; Haro:2010zz ; Haro:2010mx ; Kohri:2017iyl . The adiabatic regularization proceed the regularization through subtracting $\left<{\delta\phi}^{2}\right>^{({\rm q})}$ from $\left<{\delta\phi}^{2}\right>$ and calculates the renormalized vacuum fluctuation or the energy density in curved spacetime. Thus, the inflationary vacuum fluctuation can be written as Haro:2010zz ; Haro:2010mx ; Kohri:2017iyl

[TABLE]

where we must choose an appropriate initial vacuum and determine the mode function of $\delta\chi\left(\eta\right)$. We consider the massive non-minimally coupled case (for the details, see Ref.Haro:2010zz ; Haro:2010mx ; Kohri:2017iyl ) where the mode function in de Sitter spacetime is given by

[TABLE]

where ${H}_{\nu}^{(1,2)}\left(k\eta\right)$ are the Hankel functions with

[TABLE]

We assume the specific universe from the radiation-dominated stage to the de Sitter stage and require the matching conditions at $\eta=\eta_{0}$ to determine the Bogoliubov coefficients,

[TABLE]

From Eq. (17) the inflationary fluctuations are given as follows:

[TABLE]

where we introduce the mode cut-off $k>\sqrt{2-{M}^{2}\left(\phi\right)/H^{2}}/\left|\eta\right|\simeq\sqrt{2}/\left|\eta\right|\simeq\sqrt{2}aH$. In the limit ${\Lambda_{\textrm{cut-off}}\rightarrow\infty}$ the divergence parts exactly cancel Haro:2010zz ; Haro:2010mx ; Kohri:2017iyl ,

[TABLE]

Beyond the mode cut-off $k>\sqrt{2}aH$, the mode function does not drastically change against the evolution of the universe. In this sense the mode cut-off can be recognized as the UV cut-off of the inflationary perturbations and therefore the influences of the UV or trans-Planckian physics are sequestered as long as the Hubble parameter is much smaller than the cut-off scale, $H\ll\Lambda_{\rm UV}$.

By using the formula of the Hankel functions

[TABLE]

and the Bessel function of the first kind defined by $J_{\nu}=({H}_{\nu}^{(1)}+{H}_{\nu}^{(2)})/2$, we can obtain the expression

[TABLE]

For small $k$ modes, the the Bessel function and the Hankel function asymptotically behave as

[TABLE]

Thus, we can obtain the following expression of the mode function,

[TABLE]

For large $k$ modes, we approximate the Bogoliubov coefficients to be $\alpha_{k}\simeq 1$ and $\beta_{k}\simeq 0$ and evaluate the mode function as

[TABLE]

Thus, we can get the following expression

[TABLE]

From Eq. (28) and Eq. (30), the vacuum fluctuations are written as

[TABLE]

For late cosmic-time ($N_{\rm tot}=Ht\gg H^{2}/{M}^{2}\left(\phi\right)$), the inflationary fluctuations are approximately written as 333 The power spectrum on super-horizon scale ($\left|k\eta\right|\ll 1$) can be approximately written by Riotto:2002yw

$\displaystyle{P}_{\delta{\phi}}\left(\eta,k\right)={\left(\frac{H}{2\pi}\right)}^{2}{\left(\frac{k}{aH}\right)}^{3-2\nu}$

(32)

which has a tiny $k$-dependence, i.e scale invariance.

[TABLE]

which is consistent with the well-known results using stochastic approach (see e.g. Ref. Finelli:2008zg ; Finelli:2010sh ). Therefore, the UV divergences are sequestered and the influence of the trans-Planckian physics is negligible. Precisely, however, ${\left<{\delta\phi}^{2}\right>}^{({\rm c})}$ is mode dependent and might leave a tiny trans-Planckian imprint on the CMB at the short distance. Furthermore, the inflaton mass should be smaller than the Hubble scale if not the inflationary fluctuations are strongly suppressed Mottola:1984ar and the effective mass is formally given by the effective potential. From Eq. (16) the UV divergences are essentially sequestered and the quantum corrections are embedded in the effective potential. The quantum effects of the UV physics only changes the effective potential through the radiative corrections, but they often modify the slow roll parameter Boyanovsky:2005sh ; Sloth:2006az . In next section we construct modified effective potential from the inflationary fluctuations and discuss how the UV quantum corrections affect the potential during inflation.

III The modified effective potential during inflation

The effective potential receives the backreaction of the inflationary perturbations and UV radiative corrections. In this section we consider the modified effective potential during inflation. The inflationary expansion is determined by the effective potential and the classical slow-roll motion of the inflaton field. The effective potential is constructed by quantum loop corrections and corresponds to the effective vacuum energy density $\rho_{\rm eff}$ or pressure $P_{\rm eff}$ derived from the effective energy-momentum tensor,

[TABLE]

In the effective Friedman equations the Hubble parameter in the slow-roll approximation can be written as

[TABLE]

The slow-roll parameters $\epsilon$, $\eta$ related with the observed primordial scalar or tensor perturbations can be given as

[TABLE]

where we define $M_{\rm eff}^{2}(\phi)=V^{\prime\prime}_{\rm eff}(\phi)$ and the slow-roll approximation require $M_{\rm eff}(\phi)\ll H$.

Let us consider how to treat the quantum fluctuations during inflation using Eq. (10) and Eq. (16) again. We can regularize the UV divergences of the quantum fluctuations $\left<{\delta\phi}^{2}\right>^{({\rm q})}$ using the dimensional regularization,

[TABLE]

where:

[TABLE]

where $N_{\epsilon}=-1/{\epsilon}-\log{4\pi}-\gamma$, $\gamma$ is the Euler-Mascheroni constant and $\mu$ is the renormalization parameter. From Eq. (37) the quantum fluctuation during inflation can be written as

[TABLE]

which is almost same as flat spacetime except $\left<{\delta\phi}^{2}\right>^{({\rm c})}$. From the above relation we can get the one-loop effective potential during inflation as follows Kohri:2017iyl :

[TABLE]

where the divergences can be eliminated by the renormalization parameters, and the radiative corrections express quantum loop effects. The modified effective potential has two additional terms, non-minimal coupling terms and the back-reaction terms from the inflationary fluctuations which usually break the slow-roll conditions. Therefore, we must impose the condition on these couplings,

[TABLE]

where we assume the inflationary fluctuations are as large as the expression of Eq. (33).

Next, let us consider the quantum corrections of the massive scalar field $S$ where ${m}_{S}\gg H$ and construct the effective potential during inflation. We obtain one-loop effective potential during inflation by using $\left<{\delta\phi}^{2}\right>$ and $\left<{\delta S}^{2}\right>$,

[TABLE]

where $\left<{\delta S}^{2}\right>^{({\rm c})}$ is sufficiently suppressed and we define

[TABLE]

By using Eq. (42), we can read off the $\mu$ dependence of these couplings and the one-loop $\beta$ function of ${m}^{2}_{\phi}$ can be given as follows:

[TABLE]

where the renormalization scale $\mu$ express various phenomenological scale;

[TABLE]

Recalling that large field inflation takes the Planck field value of the inflaton and the inflaton field is larger than the Hubble scale, $\phi\sim M_{\rm P}\gg H$. Thus, the fine-tuning problem of the inflaton potential arises unless $\mu\sim\phi\ll m_{S}$. At least the slow-roll condition requires ${m}_{\phi}\ll H$ and therefore, the inflationary fluctuation or the primordial CMB perturbation are highly depend on the UV physics. In this sence, for $g^{1/2}{m}_{S}\gg H$, the slow-roll condition of the inflation violates or the inflationary fluctuation breaks the scale invariance of the spectrum of CMB perturbations (see e.g. Ref.Linde:1982uu ; Starobinsky:1982ee ; Vilenkin:1983xp ). Therefore, we can simply impose a constraint on the UV physics as follows:

[TABLE]

Then, we also get an upper bound of the interaction coupling to be $g\ll\mathcal{O}(10^{-10})$ if we take the Planck mass ${m}_{S}\sim M_{\rm P}$ and the current upper bound of the Hubble parameter Ade:2015lrj ; Ade:2015tva .

From here, let us consider the influences of the quantum corrections in more detail. The inflationary effective mass ${M}_{\rm eff}\left(\phi\right)$ including the UV radiative corrections is

[TABLE]

where we set the renormalization parameter $\mu$ adequately and the effective mass ${M}_{\rm eff}\left(\phi\right)$ takes the bare value ${M}_{\rm eff}\left(\phi\right)={m}^{2}_{\phi}$ in the limit $\phi,S,H\rightarrow 0$. If we assume $g^{1/2}{M}\left(S\right)\gg H$ the inflationary effective mass completely breaks the slow-roll conditions when

[TABLE]

which significantly changes the radiative correction terms. Thus, even if the inflaton satisfy the slow-roll conditions at the beginning of the inflation, at late time the effective mass glows and breaks the conditions. For the large field inflation $\phi\sim M_{\rm P}$ or the non-minimal coupling case the slow-roll condition require the decoupling of the inflaton sector and the massive scalar field and we can get the following conditions,

[TABLE]

From this viewpoint, the inflaton sector should be completely decoupled with any high energy physics. Thus, the Starobinsky inflation STAROBINSKY198099 or the Higgs inflation Bezrukov:2007ep ; Barvinsky:2008ia are very attractive since they do not require new physics 444 There are well-known strong correspondence Shapiro:2008sf ; Salvio:2015kka ; Calmet:2016fsr between the Starobinsky inflation ($a_{1}R^{2}$) and the Higgs inflation ($\xi_{H}H^{\dagger}H$). in comparison with any other inflation models Martin:2013tda ; Martin:2013nzq ; Martin:2013gra ; Ijjas:2013vea and furthermore, they matched with the current constraints of the CMB observations. The UV corrections of the massive scalar field $S$ drastically changes the effective potential of the inflaton and sometimes breaks the slow-roll conditions during inflation. These issues can be interpreted by using Eq. (33) and we can get the following expression of the inflationary fluctuation

[TABLE]

where we take $m_{S}\sim\Lambda_{\rm UV}$ and include quantum backreactions in Eq. (33) by hand. Thus, the quantum effects of the UV or trans-Planckian physics affects the primordial density perturbation significantly. As previously seen in Section II we can not easily read the the quantum effects of the UV or trans-Planckian physics in the inflationary perturbation since the quantum fluctuation of the flat spacetime and the curved spacetime is indistinguishable in the UV region. Constructing the effective potential, we can systemically read the trans-Planckian effects on the primordial density perturbation.

IV Conclusion and Discussion

In the present paper we discuss how trans-Planckian physics affects inflationary vacuum fluctuations and primordial density perturbations. Here, we consider the two-point correlation function $\left<{\delta\phi}^{2}\right>$ of the non-minimally coupled scalar fields and constructs effective potential. We have clearly shown that the cut-off divergence of the quantum fluctuations does not drastically change during inflation and the quantum corrections can be systemically embedded in standard effective potential. We have constructed the modified effective potential from the inflationary fluctuations using the adiabatic (WKB) regularization or approximation. The different points compared with the standard effective potential in flat spacetime are that Eq. (40) and Eq. (42) includes the backreaction terms of the inflationary fluctuations, non-minimal coupling of the gravity and the UV corrections of the massive scalar field. These terms generally breaks the slow-roll conditions and significantly affects the primordial density perturbation. We have obtained various conditions to succeed the slow-roll inflation like Eq. (41), Eq. (45) and Eq. (48) and got the conjecture $\Lambda_{\rm UV}\ll H/g^{1/2}$ where $g$ is the interaction coupling at the new physics scale $\Lambda_{\rm UV}$. The inflaton sector should be completely decoupled with the UV sector and, Starobinsky inflation or Higgs inflation are very attractive from this viewpoint.

Acknowledgments: I would like to thank Kazunori Kohri for numerous helpful discussions and collaboration.