Cosmological Spectrum of Two-Point Correlation Function from Vacuum Fluctuation of Stringy Axion Field in De Sitter Space: A Study of the Role of Quantum Entanglement
Abstract
:1. Introduction
- Field operator expansion (FOE) method with entangled state,
- Reduced density matrix formalism (RDM) with mixed state and
- Non-entangled state (NES) method.
- Q1. Why did we use three different formalisms to compute the cosmological two point correlation function?
- Q2. What is the correct physics they believe that happens in the setup of the space time?
- Q3. In those three formalisms, the physics is completely different. So which one is correct?
- Q4. We finally could only observe one possible observational consequence. So which one is correct?
- A1. We used three different formalisms to compute the cosmological two point correlation function to check the explicit role of quantum mechanical entanglement in the primordial cosmology. In these three formalisms the leading order expressions become same. However, the difference only can be found once we look into the small quantum corrections appearing in these formalisms. If the signature of quantum entanglement will be detected in near future in the observational probes of early universe, then one can explicitly rule out the possibility of appearing of NES method in the context of quantum field theory of primordial cosmology. On the other hand, if the signatures of quantum entanglement cannot be confirmed then one can strongly rely on the result obtained in the NES method. Additionally, it is important to note that these three frameworks provide us the quantum mechanical origin of quantum field theory of early universe cosmology.
- A2 and A3. From the theoretical perspective these three different formalisms have their own merit on the physical ground. If the quantum mechanical origin of the quantum correction of the primordial fluctuation is coming from the non entangled state then NES formalism is the only single option which can take care of the correct physics. On the other hand, if the quantum mechanical origin of the quantum correction of the primordial fluctuation is coming from the entangled mixed state then RDM formalism applicable to the subsystem is the most promising option which supports correct physical explanation. The last option is FOE formalism which is applicable when the quantum mechanical origin of the quantum correction of the primordial fluctuation is guided by the total entangled state (not the subsystem) then FOE formalism is useful to describe the correct physics.
- A4. It is very well known fact that at late time scale all the large scale structure is formed due to long range persistent correlation originated from the primordial quantum mechanical fluctuation in the early universe. This can only be consistently theoretically established by using FOE and RDM formalisms which supports the concept of quantum entanglement in early universe cosmology. Now RDM formalism is more theoretically consistent than the FOE method as it is based on the quantum description of the reduced subsystem. Now as far as the detection in the observation is concerned, if we can detect the quantum mechanical origin of the sub leading quantum correction in near future probes then one can explicitly very the explicit role of quantum entanglement, precisely test FOE or RDM formalism is correct. If we cannot detect the role of quantum entanglement then NES formalism will provide the correct physical explanation of the quantum origin of the sub leading correction term in the two point primordial correlation function.
- A complete quantum approach to compute the primordial power spectrum of mean square vacuum fluctuation, which is not usually followed in the context of cosmology.
- For the specific structure of the axion effective potential, we computed the explicit form of the corrections which are due to quantum effects.
- For our calculation, we used three different approaches at super horizon time scale hoping that the quantum corrections, at small and large wave number limits when confronted with observations, can select the most effective approach and the nature of quantum corrections. From the cosmological perspective we believe this is a very important step forward.
2. Wave Function of Axion in Open Chart
2.1. Background Geometry
2.2. Wave Function for Axion Using Bunch Davies Vacuum
2.3. Wave Function for Axion Using Vacua
3. Cosmological Spectrum of Quantum Vacuum Fluctuation
- Field operator expansion (FOE) method:This method is useful for entangled quantum states with the wave function of the de Sitter universe for Bunch Davies and most generalised vacua. Technically this formalism is based on the wave function which we will explicitly derive. The cosmological spectrum is characterised by the two point correlation function and their associated power spectrum. Using such entangled state in this formalism one can construct the usual density matrix for Bunch Davies and most generalised vacua.
- Reduced density matrix (RDM) formalism:This formalism is helpful for mixed quantum states and is useful for the construction of reduced density matrix in a diagonalised representation of Bunch Davies and vacua by tracing over the all possible degrees of freedom from the region R. Technically the formalism is based on the wave function which we explicitly derive.
- Non entangled state (NES) formalism:This formalism in presence of non entangled quantum state which deals with the construction of wave function in the region L in which the total universe is described. Here we also use Bunch Davies and most generalised vacua in the region L. Technically this formalism is based on the wave function which we explicitly derive in this paper.
- First of all, we trace out all contributions which belong to the R region. As a result, the required field operator is only defined in the L region. This method we use in FOE formalism where the quantum states for L and R region are entangled with each other. On the other hand, doing a partial trace over region R one can construct reduced density matrix which leads to RDM formalism. Instead, if we use the non entangled quantum state and compute the wave function solely in L region we will be lead to the NES formalism. Please note that all of these three methods are used to compute mean square vacuum fluctuation or more precisely the quantum mechanical computation of two point correlation function for axion and the associated power spectrum.
- Instead of doing the computation in basis we use a new basis , obtained by applying Bogoliubov transformation in . Consequently the field operators will act on and the FOE method is developed in this transformed basis. On the other hand, as mentioned earlier it will appear in the expression for the reduced density matrix to be used in the RDM formalism. However, in the NES formalism this transformation is not very useful since in this case the total wave function is solely described by the quantum mechanical state appearing in the L region and the corresponding Hilbert space is spanned by only which forms a complete basis.
- Furthermore, we will compute the expressions for the mean square quantum vacuum fluctuation and the corresponding cosmological power spectrum after horizon exit using all the three formalisms, i.e., FOE, RDM, and NES. We will finally consider two limiting situations: long wave length and short wave length approximation for the computation of the power spectrum.
3.1. Quantum Vacuum Fluctuation Using Field Operator Expansion (FOE) (with Entangled State)
3.1.1. Wave Function in Field Operator Expansion (FOE)
3.1.2. Two Point Correlation Function
3.2. Quantum Vacuum Fluctuation Using Reduced Density Matrix (RDM) Formalism (with Mixed State)
3.2.1. Reduced Density Matrix (RDM) Formalism
3.2.2. Two Point Correlation Function
3.3. Quantum Vacuum Fluctuation with Non Entangled State (NES)
3.3.1. Non Entangled State (NES) Formalism
3.3.2. Two Point Correlation Function
4. Summary
- We explicitly studied the power spectrum of mean squared vacuum fluctuation for axion field using the concept of quantum entanglement in de Sitter space. The effective action for the axion field, used here, has its origin from Type IIB String theory compacted to four dimensions. For our analysis, we chose two initial vacuum states, i.e., Bunch Davies and a generalised class of vacua. The power spectrum of mean squared vacuum fluctuation is computed using three distinctive formalisms: (1) Field operator expansion (FOE), (2) Reduced density matrix (RDM) and (3) Non entangled state (NES). In all three cases, the computation has been done starting with two open charts in hyperbolic manifold of de Sitter space consisting of two regions: L and R. Though the starting point is same, the construction of these three formalisms are different from each other and have their own physical significance. Each of the formalism has been discussed in text of the papers and some details of approximations for them are presented in the Appendices Appendix A, Appendix B and Appendix C. Similarities and differences from each other are presented in a table.
- In case of FOE formalism, we solve for the wave function in the region L and using this solution we compute the general expression for the mean square vacuum fluctuation and its quantum correction in terms of two point correlation function. The result is evaluated at all momentum scales. We considered two limiting approximation in the characteristic momentum scales, i.e., large wave number (small wave length in which the corresponding scale is smaller than the curvature radius of the de Sitter hyperbolic open chart) regime and small wave number (long wave length in which the corresponding scale is larger than the curvature radius of the de Sitter hyperbolic open chart) regime. We observed distinctive features in the power spectrum of of mean squared vacuum fluctuation in these two different regimes. In the large wave number (small wave length) regime we found that the leading order result for the power spectrum is consistent with the known result for observed cosmological correlation function in the super horizon time scale. The correction to the leading order result that we computed for the power spectrum can be interpreted as the sub-leading effect in the observed cosmological power spectrum. This is a strong information from the perspective of cosmological observation since such effects, possibly due to quantum entanglement of states, can play a big role to break the degeneracy of the observed cosmological power spectrum in the small wave length regime. On the other hand, in the long wave length regime we found that the power spectrum follows completely different momentum dependence in the super horizon time scale. Since in this regime and in this time scale, at present, we lack adequate observational data on power spectrum we are unable to comment on our result with observation. However, our result for the power spectrum in long wave length limit and super horizon time scale can be used as a theoretical probe to study the physical implications and its observational cosmological consequences in near future. Our result also implies that the mean square vacuum fluctuation for axion field, in super horizon time scale, gets enhanced in long wave length regime and freezes in the small wave length regime. We also observe that for a massive axion, the power spectrum is nearly scale invariant in all momentum scales. On the other hand, for massless axion we observe exact scale invariance only in large wave number (small wave length) regime and for the Bunch Davies initial quantum state. For generalised initial state, we find slight modification in the corresponding power spectrum of the mean square vacuum fluctuation. The modification factor is proportional to which is valid for all values of the parameter . It also implies that for large value of the parameter we get additional exponential suppression for the power spectrum. This information can be used to distinguish between the role of Bunch Davies vacuum () and any vacua quantum initial state during analysis of observational data.
- In RDM formalism, the wave function for the axion field is solved in L and R regions of the de Sitter open chart. This solution was used to compute the mean square vacuum fluctuation and its quantum correction for both Bunch Davies and vacuum state. Corresponding results are evaluated at all momentum scales by partially tracing out all the information from the region R. Like in the case of FOE, we considered the small and large wavelength approximations in the characteristic momentum scales and found distinct features in the corresponding power spectrum. In the small wave length regime again the leading order result, in super horizon time scales matched with known result (same as FOE). However, the sub-leading order result for the power spectrum is different from the result obtained from FOE formalism which distinguishes the two approaches. Moreover, in the long wave length regime the power spectrum has completely different momentum dependence compared to FOE formalism. We also noticed that the enhancement of mean square vacuum fluctuation for axion field, in long wave length regime, is different (slower) in nature compared to FOE formalism but the freezing in short wavelength regime is of same nature. The observation on scale invariance of power spectrum in this formalism remains similar to that in FOE formalism.
- In the last formalism, i.e., NES, the wave function of axion field is solved in the region L of the de Sitter hyperbolic open chart. With the help of this solution, t we computed the mean square vacuum fluctuation using Bunch Davies and vacuum state configuration. The corresponding result is evaluated at all momentum scales. Like the previous two cases, here also we reverted to two limiting approximations, i.e., large wave number (small wave length ) regime and small wave number (long wave length) regime. We again observed distinctive behaviour in the power spectrum in these two different regimes. In the large wave number (small wave length) regime, the leading order result for power spectrum matches with the known result for observed cosmological correlation function just as the cases of FOE and RDM formalism. However, the sub-leading order result s completely different FOE as well as RDM formalism. Thus, it is the sub-leading terms which distinguish these formalisms from each other and they can be confronted with future observational data. On the other hand, in the small wave number (long wave length) regime, even the leading order result for the power spectrum differs, in momentum dependence, compared to the result obtained from FOE and RDM formalism. Also the nature of enhancement of the mean square vacuum fluctuation in NES formalism is found to be different from that in FOE and RDM formalism but the nature of freezing and the observation on scale invariance of power spectrum remains same in all the three cases.
- For completeness, we discuss the actual reason for the results obtained for the power spectra from quantum entangled state as appearing in FOE formalism and the mixed state which is used to construct the RDM formalism. To do so, we consider two subsystems, L and R using which one can construct the quantum mechanical state vector of axion field as . In our computation, these subsystems are defined in the region L and R respectively in the de Sitter hyperbolic open chart. Now using this state vector of axion field we can define the density matrix as:
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Quantum Correction to the Power Spectrum in FOE Formalism
Appendix A.1. For Large Wave Number
Appendix A.2. For Small Wave Number
Appendix B. Quantum Correction to the Power Spectrum in RDM Formalism
Appendix B.1. For Large Wave Numbers
Appendix B.2. For Small Wave Number
Appendix C. Quantum Correction to the Power Spectrum in NES Formalism
Appendix C.1. For Large Wave Numbers
Appendix C.2. For Small Wave Number
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1. | It is important to note that, by the term Bunch-Davies vacuum here we actually pointing towards the well known Euclidean vacuum state which is actually a false vacuum state in quantum field theory and commonly used to fix the initial quantum condition of our universe in terms of quantum mechanical state or the wave function of the universe. |
2. | |
3. | Here the wave number p mimics the role of principal quantum number in the de Sitter hyperbolic open chart which is continuous and lying within the range . The other quantum numbers m (azimuthal) and l (orbital) play no significant role in the final result as the expression for the power spectrum for mean square vacuum fluctuation only depends on the quantum number p. |
4. | Here q is any positive odd integer. |
5. | Here it is important to note the expression for the time dependent function for (where q is any positive odd integer) in all cases are same. The only difference is appearing in the expression for the power spectrum. For case the power spectrum is scale invariant exactly. However, for the other values of the power spectrum is not scale invariant and small deviation from the scale invariant feature can be observed easily. |
Feuatures | FOE | RDM | NES |
---|---|---|---|
Wave function | Here we solve the wave function in L region of dS space. | Here we solve the wave function in L and R region of dS space. | Here we only solve the wave function in L region of dS space. |
Quantum state | Here we deal with entangled quantum state. | Here we deal with mixed quantum state. | Here we deal with non-entangled quantum state. |
Quantum number dependence | Power spectrum is only dependent on SO(1,3) quantum number p and independent on l,m. | Power spectrum is only dependent on SO(1,3) quantum number p and independent on l,m. | Power spectrum is only dependent on SO(1,3) quantum number p and independent on l,m. |
Time scale for computation | Analysis is performed on superhorizon time scale. | Analysis is performed on superhorizon time scale. | Analysis is performed on superhorizon time scale. |
Power spectrum spectrum at large wave number | Leading order term is and the next order effects are different from RDM and NES for massless axion ( ). | Leading order term is and the next order effects are different from FOE and NES for massless axion ( ). | Leading order term is and the next order effects are different from FOE and RDM for massless axion ( ). |
Power spectrum at small at small wave number | >Leading order term is and the next order effects are different from RDM and NES for massless axion( ). | Leading order term is and the next order effects are different from FOE and NES for massless axion( ). | Leading order term is and the next order effects are different from FOE and RDM for massless axion( ). |
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Choudhury, S.; Panda, S. Cosmological Spectrum of Two-Point Correlation Function from Vacuum Fluctuation of Stringy Axion Field in De Sitter Space: A Study of the Role of Quantum Entanglement. Universe 2020, 6, 79. https://doi.org/10.3390/universe6060079
Choudhury S, Panda S. Cosmological Spectrum of Two-Point Correlation Function from Vacuum Fluctuation of Stringy Axion Field in De Sitter Space: A Study of the Role of Quantum Entanglement. Universe. 2020; 6(6):79. https://doi.org/10.3390/universe6060079
Chicago/Turabian StyleChoudhury, Sayantan, and Sudhakar Panda. 2020. "Cosmological Spectrum of Two-Point Correlation Function from Vacuum Fluctuation of Stringy Axion Field in De Sitter Space: A Study of the Role of Quantum Entanglement" Universe 6, no. 6: 79. https://doi.org/10.3390/universe6060079